ON
THE
ESSENTIAL
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
IN
TERMS
OF
LOGICAL
AND
“∧”/LOGICAL
OR
“∨”
RELATIONS:
REPORT
ON
THE
OCCASION
OF
THE
PUBLICATION
OF
THE
FOUR
MAIN
PAPERS
ON
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
Shinichi
Mochizuki
September
2023
Abstract.
The
main
goal
of
the
present
paper
is
to
give
a
detailed
exposition
of
the
essential
logical
structure
of
inter-universal
Teichmüller
theory
from
the
point
of
view
of
the
Boolean
operators
—
such
as
the
logical
AND
“∧”
and
logical
OR
“∨”
operators
—
of
propositional
calculus.
This
essential
logical
structure
of
inter-universal
Teichmüller
theory
may
be
summarized
symbolically
as
follows:
A
∧
B
=
˙
B
2
∨
˙
...
)
A
∧
(B
1
∨
˙
B
2
∨
˙
...
∨
˙
B
1
∨
˙
B
2
∨
˙
...
)
A
∧
(B
1
∨
.
.
.
=⇒
—
where
˙
denotes
the
Boolean
operator
exclusive-OR,
i.e.,
“XOR”;
·
the
“
∨”
·
A,
B,
B
1
,
B
2
,
B
1
,
B
2
,
denote
various
propositions;
·
the
logical
AND
“∧’s”
correspond
to
the
Θ-link
of
inter-universal
Te-
ichmüller
theory
and
are
closely
related
to
the
multiplicative
structures
of
the
rings
that
appear
in
the
domain
and
codomain
of
the
Θ-link;
˙
·
the
logical
XOR
“
∨’s”
correspond
to
various
indeterminacies
that
arise
mainly
from
the
log-Kummer-correspondence,
i.e.,
from
sequences
of
it-
erates
of
the
log-link
of
inter-universal
Teichmüller
theory,
which
may
be
thought
of
as
a
device
for
constructing
additive
log-shells.
˙
This
sort
of
concatenation
of
logical
AND
“∧’s”
and
logical
XOR
“
∨’s”
is
reminiscent
of
the
well-known
description
of
the
“carry-addition”
operation
on
Teichmüller
˙
representatives
of
the
truncated
Witt
ring
Z/4Z
in
terms
of
Boolean
addition
“
∨”
and
Boolean
multiplication
“∧”
in
the
field
F
2
and
may
be
regarded
as
a
sort
of
“Boolean
intertwining”
that
mirrors,
in
a
remarkable
fashion,
the
“arith-
metic
intertwining”
between
addition
and
multiplication
in
number
fields
and
local
fields,
which
is,
in
some
sense,
the
main
object
of
study
in
inter-universal
Te-
ichmüller
theory.
One
important
topic
in
this
exposition
is
the
issue
of
“redundant
copies”,
i.e.,
the
issue
of
how
the
arbitrary
identification
of
copies
of
isomorphic
mathematical
objects
that
appear
in
the
various
constructions
of
inter-universal
Te-
ichmüller
theory
impacts
—
and
indeed
invalidates
—
the
essential
logical
structure
of
inter-universal
Teichmüller
theory.
This
issue
has
been
a
focal
point
of
funda-
mental
misunderstandings
and
entirely
unnecessary
confusion
concerning
inter-universal
Teichmüller
theory
in
certain
sectors
of
the
mathematical
community.
The
exposition
of
the
topic
of
“redundant
copies”
makes
use
of
many
interesting
elementary
examples
from
the
history
of
mathematics.
Typeset
by
AMS-TEX
1
2
SHINICHI
MOCHIZUKI
Contents:
Introduction
§1.
Summary
of
non-mathematical
aspects
for
non-specialists
§1.1.
Publication
of
[IUTchI-IV]
§1.2.
Redundancy
assertions
of
the
“redundant
copies
school”
(RCS)
§1.3.
Qualitative
assessment
of
assertions
of
the
RCS
§1.4.
The
importance
of
extensive,
long-term
interaction
§1.5.
The
historical
significance
of
detailed,
explicit,
accessible
records
§1.6.
The
importance
of
further
dissemination
§1.7.
The
notion
of
an
“expert”
§1.8.
Fabricated
versions
spawn
fabricated
dramas
§1.9.
Geographical
vs.
mathematical
proximity
§1.10.
Mathematical
intellectual
property
rights
§1.11.
Social
mirroring
of
mathematical
logical
structure
§1.12.
Computer
verification,
mathematical
dialogue,
and
developmental
reconstruction
§2.
Elementary
mathematical
aspects
of
“redundant
copies”
§2.1.
The
history
of
limits
and
integration
§2.2.
Derivatives
and
integrals
§2.3.
Line
segments
vs.
loops
§2.4.
Logical
AND
“∧”
vs.
logical
OR
“∨”
§3.
The
logical
structure
of
inter-universal
Teichmüller
theory
§3.1.
One-dimensionality
via
identification
of
RCS-redundant
copies
§3.2.
RCS-redundancy
of
Frobenius-like/étale-like
versions
of
objects
§3.3.
RCS-redundant
copies
in
the
domain/codomain
of
the
log-link
§3.4.
RCS-redundant
copies
in
the
domain/codomain
of
the
Θ-link
§3.5.
Gluings,
indeterminacies,
and
pilot
discrepancy
§3.6.
Chains
of
logical
AND
relations
§3.7.
Poly-morphisms
and
logical
AND
relations
§3.8.
Inter-universality
and
logical
AND
relations
§3.9.
Passage
and
descent
to
underlying
structures
§3.10.
Detailed
description
of
the
chain
of
logical
AND
relations
§3.11.
The
central
importance
of
the
log-Kummer-correspondence
List
of
Examples:
1.5.1.
Irrationality,
impartiality,
and
the
Voodoo
Hypothesis
1.5.2.
The
internet/mass
media
as
an
apple
of
discord
1.9.1.
The
insufficiency
of
geographical
proximity
1.9.2.
The
remarkable
potency
of
mathematical
proximity
1.10.1.
The
Pythagorean
Theorem
1.12.1.
Explicit
parametrization
of
Pythagorean
triples
2.1.1.
False
contradiction
in
the
theory
of
integration
2.2.1.
Symmetry
properties
of
derivatives
2.3.1.
Endpoints
of
an
oriented
line
segment
2.3.2.
Gluing
of
adjacent
oriented
line
segments
2.4.1.
“∧”
vs.
“∨”
for
adjacent
oriented
line
segments
2.4.2.
Differentials
on
oriented
line
segments
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
3
2.4.3.
Representation
via
subgroup
indices
of
“∧”
vs.
“∨”
2.4.4.
Logical
“∧/∨”
vs.
“narrative
∧/∨”
2.4.5.
Numerical
representation
of
“∧”
vs.
“∨”
2.4.6.
Carry
operations
in
arithmetic,
geometry,
and
Boolean
logic
2.4.7.
The
projective
line
as
a
gluing
of
ring
schemes
along
a
multiplicative
group
scheme
2.4.8.
Gluings
of
rings
along
multiplicative
monoids
3.1.1.
Elementary
models
of
gluings
and
intertwinings
3.2.1.
Global
multiplicative
subspaces
and
bounds
on
heights
3.2.2.
Coricity,
symmetry,
and
commutativity
properties
of
the
log-theta-lattice
3.3.1.
Classical
complex
Teichmüller
theory
3.3.2.
The
Jacobi
identity
for
the
classical
theta
function
3.3.3.
Theta
functions
and
multiplicative
structures
3.5.1.
Bounded
nature
of
log-shell
automorphism
indeterminacies
3.5.2.
Examples
of
gluings
3.8.1.
Inevitability
of
inner
automorphism
indeterminacies
3.8.2.
Inter-universality
and
the
structure
of
(Θ
±ell
NF-)Hodge
theaters
3.8.3.
Truncated
vs.
profinite
Kummer
theory
and
compatibility
with
the
p-adic
logarithm
3.8.4.
Symmetrizing
isomorphisms,
truncatibility,
and
the
log-Kummer-
correspondence
3.9.1.
Categories
of
open
subschemes
3.10.1.
Symmetries
as
a
fundamental
non-formal
aspect
of
gluings
3.10.2.
Chains
of
logical
AND
relations
via
commutative
diagrams
Introduction
In
the
present
paper,
we
give
a
detailed
exposition
of
the
essential
logical
structure
of
inter-universal
Teichmüller
theory
in
terms
of
elementary
Boolean
operators
such
as
logical
AND
“∧”
and
logical
OR
“∨”.
One
important
topic
in
this
exposition
is
the
issue
of
“redundant
copies”,
i.e.,
the
issue
of
how
the
arbi-
trary
identification
of
copies
of
isomorphic
mathematical
objects
that
appear
in
the
various
constructions
of
inter-universal
Teichmüller
theory
impacts
—
and
in-
deed
invalidates
—
the
essential
logical
structure
of
inter-universal
Teichmüller
the-
ory.
This
issue
has
been
a
focal
point
of
fundamental
misunderstandings
and
entirely
unnecessary
confusion
concerning
inter-universal
Teichmüller
theory
in
certain
sectors
of
the
mathematical
community
[cf.
the
discussion
of
Examples
2.4.5,
2.4.7,
2.4.8].
We
begin,
in
§1,
by
reporting
on
various
non-mathematical
aspects
of
the
situa-
tion
surrounding
inter-universal
Teichmüller
theory,
such
as
the
issue
of
“redundant
copies”.
Perhaps
the
most
central
portion
of
this
discussion
of
non-mathematical
as-
pects
of
the
situation
surrounding
inter-universal
Teichmüller
theory
concerns
the
long-term,
historical
importance
of
producing
detailed,
explicit,
mathemati-
cally
substantive,
and
readily
accessible
written
documentation
of
the
essential
logical
structure
of
the
issues
under
debate
[cf.
§1.5].
Such
written
documentation
of
the
essential
logical
structure
of
the
issues
under
debate
is
especially
important
in
situations
such
as
the
situation
that
has
arisen
surrounding
inter-universal
Te-
ichmüller
theory,
in
which
the
proliferation
of
logically
unrelated
fabricated
4
SHINICHI
MOCHIZUKI
versions
of
the
theory
has
led
to
fundamental
misunderstandings
and
entirely
un-
necessary
confusion
concerning
inter-universal
Teichmüller
theory
in
certain
sectors
of
the
mathematical
community
that
are
deeply
detrimental
to
the
operational
normalcy
of
the
field
of
mathematics
[cf.
§1.3,
§1.8,
§1.10,
§1.11,
§1.12].
This
dis-
cussion
in
§1
is
supplemented
by
various
interesting
historical
examples
related
to
the
irrationality
of
square
roots
of
prime
numbers
[cf.
Example
1.5.1],
the
Pythagorean
Theorem
[cf.
Example
1.10.1],
and
Pythagorean
triples
[cf.
Example
1.12.1].
We
then
proceed,
in
§2,
to
discuss
elementary
aspects
of
the
mathematics
surrounding
the
essential
logical
structure
of
inter-universal
Teichmüller
theory.
Our
discussion
of
these
elementary
aspects,
which
concerns
mathematics
at
the
ad-
vanced
undergraduate
or
beginning
graduate
level
and
does
not
require
any
advanced
knowledge
of
anabelian
geometry
or
inter-universal
Teichmüller
theory,
focuses
on
the
close
relationship
between
·
integration
and
differentiation
on
—
i.e.,
so
to
speak,
the
“differential
geometry”
of
—
the
real
line
[cf.
§2.1,
§2.2,
as
well
as
Example
2.4.2],
·
the
geometry
of
adjacent
closed
intervals
of
the
real
line
and
the
loops
that
arise
by
identifying
various
closed
subspaces
of
such
closed
intervals
[cf.
§2.3;
Example
2.4.1],
and
·
Boolean
operators
such
as
logical
AND
“∧”
and
logical
OR
“∨”
[cf.
§2.4].
One
important
unifying
theme
that
relates
these
seemingly
disparate
topics
is
the
theme
of
“carry
operations”,
which
appear
in
the
various
arithmetic,
geomet-
ric
[i.e.,
“gluing”],
and
Boolean-logical
situations
discussed
in
§2
[cf.
Example
2.4.6].
On
the
other
hand,
from
the
point
of
view
of
arithmetic
geometry,
the
discus-
sion
of
the
projective
line
as
a
gluing
of
ring
schemes
along
a
multiplicative
group
scheme
given
in
Example
2.4.7
yields
a
remarkably
elementary
qualitative
model/analogue
of
the
essential
logical
structure
surrounding
the
gluing
given
by
the
Θ-link
in
inter-
universal
Teichmüller
theory.
Moreover,
over
the
complex
numbers,
this
example
of
the
projective
line
—
i.e.,
which
may
be
visualized
as
a
sphere
—
leads
to
an
inter-
esting
analogy
between
the
well-known
[e.g.,
especially,
in
a
cartographic
context!]
metric/geodesic
geometry
of
the
sphere
with
the
multiradial
representa-
tion
of
the
Θ-pilot
in
inter-universal
Teichmüller
theory
[cf.
Example
2.4.7,
(v)].
This
example
of
the
projective
line
discussed
in
Example
2.4.7
may
be
understood
as
occupying
a
special
role
in
the
exposition
of
the
present
paper
in
light
of
the
fact
that
it
is
more
directly
related
to
scheme-theoretic
arithmetic
geometry
than
the
previously
mentioned
examples
and
leads
naturally
to
the
subsequent
ring-/monoid-theoretic
Example
2.4.8,
which
may
literally
be
regarded,
i.e.,
in
a
much
more
rigorous,
technical
sense,
as
a
sort
of
miniature
qualitative
model
—
that
is
to
say,
so
to
speak,
a
sort
of
“preview”
—
of
the
gluing
constituted
by
the
Θ-link
of
inter-universal
Teichmüller
theory.
Finally,
this
example
of
the
projec-
tive
line
is
also
of
interest
in
light
of
the
remarkable
parallels
between
the
issue
of
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
5
“redundant
copies”
in
the
context
of
inter-universal
Teichmüller
theory
and
the
well-known
19-th
century
“algebraic
truths”
versus
“geometric
fantasies”
dis-
pute
between
Weierstrass
and
Riemann
concerning
approaches
to
complex
function
theory
[cf.
the
discussion
of
§1.5].
The
preparatory
topics
of
§2
lead
naturally
to
the
detailed
exposition
of
the
essential
logical
structure
of
inter-universal
Teichmüller
theory
given
in
§3.
From
a
strictly
rigorous
point
of
view,
this
exposition
assumes
a
substantial
level
of
knowl-
edge
and
understanding
of
the
technicalities
of
inter-universal
Teichmüller
theory
[which
are
surveyed,
for
instance,
in
[Alien]],
although
the
essential
mathematical
content
of
most
of
the
issues
discussed
may
in
fact
be
understood
at
the
level
of
the
elementary
considerations
discussed
in
§2.
The
essential
logical
structure
of
inter-universal
Teichmüller
theory
may
be
represented
symbolically
as
follows:
A
∧
B
=⇒
A
∧
(B
1
∨
˙
B
2
∨
˙
.
.
.
)
A
∧
(B
1
∨
˙
B
2
∨
˙
.
.
.
∨
˙
B
1
∨
˙
B
2
∨
˙
.
.
.
)
=⇒
A
∧
(B
1
∨
˙
B
2
∨
˙
.
.
.
∨
˙
B
1
∨
˙
B
2
∨
˙
.
.
.
∨
˙
B
1
∨
˙
B
2
∨
˙
.
.
.
)
=
..
.
˙
˙
denotes
the
Boolean
—
cf.
the
discussion
of
(∧(
∨)-Chn)
in
§3.10.
[Here,
“
∨”
operator
exclusive-OR,
i.e.,
“XOR”.]
Indeed,
§3
is
devoted,
for
the
most
part,
to
giving
a
detailed
exposition
of
various
aspects
of
this
symbolic
representation,
such
as
the
following:
·
the
logical
AND
“∧’s”
in
the
above
display
may
be
understood
as
corresponding
to
the
Θ-link
of
inter-universal
Teichmüller
theory
and
are
closely
related
to
the
multiplicative
structures
of
the
rings
that
appear
in
the
domain
and
codomain
of
the
Θ-link;
˙
·
the
logical
XOR
“
∨’s”
in
the
above
display
may
be
understood
as
corresponding
to
various
indeterminacies
that
arise
mainly
from
the
log-
Kummer-correspondence,
i.e.,
from
sequences
of
iterates
of
the
log-
link
of
inter-universal
Teichmüller
theory,
which
may
be
thought
of
as
a
device
for
constructing
additive
log-shells.
˙
This
appearance
of
logical
AND
“∧’s”
and
logical
XOR
“
∨’s”
is
of
interest
in
that
it
is
reminiscent
of
the
well-known
description
of
the
“carry-addition”
operation
on
Teichmüller
representatives
of
the
truncated
Witt
ring
Z/4Z
in
terms
of
Boolean
˙
and
Boolean
multiplication
“∧”
in
the
field
F
2
and
may
be
re-
addition
“
∨”
garded
as
a
sort
of
“Boolean
intertwining”
that
mirrors,
in
a
remarkable
fashion,
the
“arithmetic
intertwining”
between
addition
and
multiplication
in
num-
ber
fields
and
local
fields,
which
is,
in
some
sense,
the
main
object
of
study
in
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
Example
2.4.6,
(iii);
the
discussion
surrounding
(TrHrc)
in
§3.10].
The
above
symbolic
representation
of
the
essential
logical
structure
of
inter-universal
Teichmüller
theory
arises
naturally
from
considerations
concerning
such
key
topics
as
·
the
coricity/symmetry/commutativity
properties
of
the
log-theta-
lattice
[cf.
Example
3.2.2]
and
the
closely
related
significance
of
working
with
both
Frobenius-like
and
étale-like
objects
[cf.
§3.2];
6
SHINICHI
MOCHIZUKI
·
the
closely
intertwined
properties
of
theta
functions
and
[p-adic/archi-
medean]
logarithms
[cf.
Examples
3.3.2,
3.3.3,
3.8.3,
3.8.4]
in
the
con-
text
of
the
log-Kummer-correspondence
[cf.
§3.3;
§3.11;
Examples
3.2.1,
3.3.1];
·
generalities
concerning
gluings
[cf.
§3.1,
§3.4,
§3.5,
§3.10;
Examples
3.1.1,
3.5.2,
3.10.1,
3.10.2];
·
generalities
concerning
indeterminacies
[cf.
§3.5,
§3.6,
§3.7;
Example
3.5.1];
·
generalities
concerning
inter-universality
[cf.
§3.8;
Examples
3.8.1,
3.8.2,
3.8.3,
3.8.4];
·
generalities
concerning
descent
to
underlying
structures
[cf.
§3.9,
§3.10,
§3.11].
Acknowledgements
The
present
paper
benefited
substantially
from
numerous
discussions
with
Ben-
jamin
Collas,
Ivan
Fesenko,
Yuichiro
Hoshi,
Fumiharu
Kato,
Emmanuel
Lepage,
Arata
Minamide,
Wojciech
Porowski,
Mohamed
Saı̈di,
Fucheng
Tan,
and
Shota
Tsujimura.
The
author
would
like
to
express
his
appreciation
to
all
of
these
math-
ematicians
for
the
time
and
effort
that
they
contributed
to
these
discussions.
Section
1:
Summary
of
non-mathematical
aspects
for
non-specialists
We
begin
with
an
overall
summary
of
non-mathematical
aspects
of
the
situation
surrounding
[IUTchI-IV],
which
may
be
of
interest
to
both
non-mathematicians
and
mathematicians.
We
also
refer
to
[FsADT],
[FKvid],
[FsDss],
[FsPio]
for
a
discussion
of
various
aspects
of
this
situation
from
slightly
different
points
of
view.
§1.1.
Publication
of
[IUTchI-IV]
The
four
main
papers
[IUTchI-IV]
on
inter-universal
Teichmüller
theory
(IUT)
were
accepted
for
publication
in
the
Publications
of
the
Research
Institute
for
Mathematical
Sciences
(PRIMS)
on
February
5,
2020.
This
was
announced
at
an
online
video
news
conference
held
at
Kyoto
University
on
April
3,
2020.
The
four
papers
were
subsequently
published
in
several
special
volumes
of
PRIMS,
a
leading
international
journal
in
the
field
of
mathematics
with
a
distinguished
history
dating
back
over
half
a
century.
The
refereeing
for
these
Special
Volumes
was
overseen
by
an
Editorial
Board
for
the
Special
Volumes
chaired
by
Professors
Masaki
Kashiwara
and
Akio
Tamagawa.
[Needless
to
say,
as
the
author
of
these
four
papers,
I
was
completely
excluded
from
the
activities
of
this
Editorial
Board
for
the
Special
Volumes.]
Professor
Kashiwara,
a
professor
emeritus
at
RIMS,
Kyoto
University,
is
a
global
leader
in
the
fields
of
algebraic
analysis
and
representation
theory.
Professor
Tamagawa,
currently
a
professor
at
RIMS,
Kyoto
University,
is
a
leading
pioneer
in
the
field
of
anabelian
geometry
and
related
research
in
arithmetic
geometry.
Here,
it
should
be
noted
that,
to
a
substantial
extent,
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
7
inter-universal
Teichmüller
theory
arose
as
an
extension/application
—
developed
by
the
author
in
the
highly
mathematically
stimulating
en-
vironment
at
RIMS,
Kyoto
University,
over
the
course
of
roughly
two
decades
[i.e.,
1992
-
2012]
—
of
precisely
the
sort
of
anabelian
geome-
try
that
was
pioneered
by
Tamagawa.
It
is
for
this
reason
that
PRIMS
stood
out
among
mathematics
journals
worldwide
as
the
most
appropriate
—
i.e.,
in
the
sense
of
being
by
far
the
most
[and
indeed
perhaps
the
only
truly]
technically
qualified
—
journal
for
the
task
of
refereeing
and
publishing
the
four
papers
[IUTchI-IV]
on
inter-universal
Teichmüller
theory.
Both
Professors
Kashiwara
and
Tamagawa
have
an
outstandingly
high
inter-
national
reputation,
built
up
over
distinguished
careers
that
span
several
decades.
It
is
entirely
inconceivable
that
any
refereeing
process
overseen
by
these
mathemati-
cians
might
be
conducted
relative
to
anything
less
than
the
highest
mathematical
standards,
free
of
any
inappropriate
non-mathematical
considerations.
In
an
arti-
cle
in
the
Asahi
Shimbun
[a
major
Japanese
newspaper]
published
shortly
after
the
announcement
of
April
3,
2020,
Professor
Tamagawa
is
quoted
as
saying
that
he
has
“100
percent
confidence
in
the
refereeing”
that
was
done
for
the
four
papers
[IUTchI-IV].
In
another
article
in
the
Asahi
Shimbun
[also
published
shortly
after
the
an-
nouncement
of
April
3,
2020],
Professors
Shigefumi
Mori,
a
professor
emeritus
at
RIMS,
Kyoto
University,
and
Nobushige
Kurokawa,
a
professor
emeritus
at
the
Tokyo
Institute
of
Technology,
express
their
expectations
about
the
possibility
of
applying
inter-universal
Teichmüller
theory
to
other
unsolved
problems
in
number
theory.
Subsequent
to
these
developments
in
2020,
a
sequel
[ExpEst]
to
the
four
original
papers
[IUTchI-IV]
on
inter-universal
Teichmüller
theory
was
accepted
for
publica-
tion
in
the
Kodai
Mathematical
Journal
in
September
2021.
This
sequel
[ExpEst]
concerns
explicit
numerical
estimates
in
inter-universal
Teichmüller
theory
and
con-
tains,
in
particular,
a
new
proof
of
Fermat’s
Last
Theorem.
In
particular,
the
results
proven
in
the
four
original
papers
[IUTchI-IV]
on
inter-universal
Teichmüller
theory,
as
well
as
the
sequel
[ExpEst],
may
now
be
quoted
in
the
mathematical
literature
as
results
proven
in
papers
that
have
been
published
in
leading
international
journals
in
the
field
of
mathematics
after
under-
going,
in
the
case
of
the
four
original
papers
[IUTchI-IV],
an
exceptionally
thorough
[seven
and
a
half
year
long]
refereeing
process.
§1.2.
Redundancy
assertions
of
the
“redundant
copies
school”
(RCS)
Unfortunately,
it
has
been
brought
to
my
attention
that,
despite
the
develop-
ments
discussed
in
§1.1,
fundamental
misunderstandings
concerning
the
math-
ematical
content
of
inter-universal
Teichmüller
theory
persist
in
certain
sectors
of
the
mathematical
community.
These
misunderstandings
center
around
a
certain
oversimplification
—
which
is
patently
flawed,
i.e.,
leads
to
an
immediate
contradic-
tion
—
of
inter-universal
Teichmüller
theory.
This
oversimplified
version
of
inter-
universal
Teichmüller
theory
is
based
on
assertions
of
redundancy
concerning
8
SHINICHI
MOCHIZUKI
various
multiple
copies
of
certain
mathematical
objects
that
appear
in
inter-
universal
Teichmüller
theory.
In
the
present
paper,
I
shall
refer
to
the
school
of
thought
[i.e.,
in
the
sense
of
a
“collection
of
closely
interrelated
ideas”]
constituted
by
these
assertions
as
the
“RCS”,
i.e.,
“redundant
copies
school
[of
thought]”.
One
fundamental
reason
for
the
use
of
this
term
“RCS”
[i.e.,
“redundant
copies
school
[of
thought]”]
in
the
present
paper,
as
opposed
to
proper
names
of
math-
ematicians,
is
to
emphasize
the
importance
of
concentrating
on
mathematical
content,
as
opposed
to
non-mathematical
—
i.e.,
such
as
social,
political,
or
psy-
chological
—
aspects
or
interpretations
of
the
situation.
Thus,
in
a
word,
the
central
assertions
of
the
RCS
may
be
summarized
as
follows:
Various
multiple
copies
of
certain
mathematical
objects
in
inter-universal
Teichmüller
theory
are
redundant
and
hence
may
be
identified
with
one
another.
On
the
other
hand,
once
one
makes
such
identifications,
one
obtains
an
immediate
contradiction.
In
the
present
paper,
I
shall
refer
to
redundancy
in
the
sense
of
the
assertions
of
the
RCS
as
“RCS-redundancy”,
to
the
identifications
of
RCS-redundant
copies
that
appear
in
the
assertions
of
the
RCS
as
“RCS-identifications”,
and
to
the
over-
simplified
version
of
inter-universal
Teichmüller
theory
obtained
by
implementing
the
RCS-identifications
as
“RCS-IUT”.
As
discussed
in
[Rpt2018]
[cf.,
especially,
[Rpt2018],
§18],
there
is
absolutely
no
doubt
that
RCS-IUT
is
indeed
a
meaningless
and
absurd
theory
that
leads
immediately
to
a
contradiction.
A
more
technical
discussion
of
this
contradiction,
in
the
language
of
inter-universal
Teichmüller
theory,
is
given
in
§3.1
below,
while
digested
versions
in
more
elemen-
tary
language
of
the
technical
discussion
of
§3
may
be
found
in
Examples
2.4.5,
2.4.7,
2.4.8,
below.
Rather,
the
fundamental
misunderstandings
underlying
the
RCS
lie
in
the
as-
sertions
of
RCS-redundancy.
The
usual
sense
of
the
word
“redundant”
suggests
that
there
should
be
some
sort
of
equivalence,
or
close
logical
relationship,
between
the
original
version
of
the
theory
[i.e.,
IUT]
and
the
theory
obtained
[i.e.,
RCS-IUT]
by
implementing
the
RCS-identifications
of
RCS-redundant
objects.
In
fact,
however,
implementing
the
RCS-identifications
of
RCS-redundant
objects
radi-
cally
alters/invalidates
the
essential
logical
structure
of
IUT
in
such
a
fundamental
way
that
it
seems
entirely
unrealistic
to
verify
any
sort
of
“close
logical
relationship”
between
IUT
and
RCS-IUT.
A
more
technical
discussion
of
the
three
main
types
of
RCS-redundancy/RCS-
identification
—
which
we
refer
to
as
“(RC-FrÉt)”,
“(RC-log)”,
and
“(RC-Θ)”
—
is
given,
in
the
language
of
inter-universal
Teichmüller
theory,
in
§3.2,
§3.3,
§3.4,
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
9
below.
In
fact,
however,
the
essential
mathematical
content
of
these
three
main
types
of
RCS-redundancy/RCS-identification
is
entirely
elementary
and
lies
well
within
the
framework
of
undergraduate-level
mathematics.
A
discussion
of
this
essentially
elementary
mathematical
content
is
given
in
§2.3,
§2.4
below
[cf.,
especially,
Examples
2.4.5,
2.4.7,
2.4.8].
One
important
consequence
of
the
technical
considerations
discussed
in
§3
below
is
the
following:
from
the
point
of
view
of
the
logical
relationships
between
various
as-
sertions
of
the
RCS,
the
most
fundamental
type
of
RCS-redundancy
is
(RC-Θ).
That
is
to
say,
(RC-Θ)
may
be
understood
as
the
logical
cornerstone
of
the
various
assertions
of
the
RCS.
§1.3.
Qualitative
assessment
of
assertions
of
the
RCS
As
discussed
in
detail
in
§3.4
below
[cf.
also
§2.3,
§2.4],
implementing
the
logical
cornerstone
RCS-identification
of
(RC-Θ)
completely
invalidates
the
crucial
logical
AND
“∧”
property
satis-
fied
by
the
Θ-link
—
a
property
that
underlies
the
entire
logical
struc-
ture
of
inter-universal
Teichmüller
theory.
In
particular,
understanding
the
issue
of
how
the
RCS
treats
this
fundamental
conflict
between
the
RCS-identification
of
(RC-Θ)
and
the
crucial
∧-property
of
the
Θ-link
is
central
to
the
issue
of
assessing
the
assertions
of
the
RCS.
In
March
2018,
discussions
were
held
at
RIMS
with
two
adherents
of
the
RCS
concerning,
in
particular,
(RC-Θ)
[cf.
[Rpt2018],
[Dsc2018]].
Subsequent
to
these
discussions,
after
a
few
e-mail
exchanges,
these
two
adherents
of
the
RCS
informed
me
via
e-mail
in
August
2018
—
in
response
to
an
e-mail
that
I
sent
to
them
in
which
I
stated
that
I
was
prepared
to
continue
discussing
inter-universal
Teichmüller
the-
ory
with
them,
but
that
I
had
gotten
the
impression
that
they
were
not
interested
in
continuing
these
discussions
—
that
indeed
they
were
not
interested
in
continuing
these
discussions
concerning
inter-universal
Teichmüller
theory.
In
the
same
e-mail,
I
also
stated
that
perhaps
it
might
be
more
productive
to
continue
these
discussions
of
inter-universal
Teichmüller
theory
via
different
participants
[i.e.,
via
“represen-
tatives”
of
the
two
sides]
and
encouraged
them
to
suggest
possible
candidates
for
doing
this,
but
they
never
responded
to
this
portion
of
my
e-mail.
[Incidentally,
it
should
be
understood
that
I
have
no
objection
to
making
these
e-mail
messages
public,
but
will
refrain
from
doing
so
in
the
absence
of
explicit
permission
from
the
two
recipients
of
the
e-mails.]
Since
March
2018,
I
have
spent
a
tremendous
amount
of
time
discussing
the
fundamental
“(RC-Θ)
vs.
∧-property”
conflict
mentioned
above
with
quite
a
number
of
mathematicians.
Moreover,
during
the
years
following
the
March
2018
discussions,
many
mathematicians
[including
myself!]
with
whom
I
have
been
in
contact
have
devoted
a
quite
substantial
amount
of
time
and
effort
to
analyzing
and
discussing
certain
10pp.
manuscripts
written
by
adherents
of
the
RCS
[cf.,
especially,
the
discussion
of
the
final
page
and
a
half
of
the
files
“[SS2018-05]”,
10
SHINICHI
MOCHIZUKI
“[SS2018-08]”
available
at
the
website
[Dsc2018]]
—
indeed
to
such
an
extent
that
by
now,
many
of
us
can
cite
numerous
key
passages
in
these
manuscripts
by
memory.
More
recently,
one
mathematician
with
whom
I
have
been
in
contact
has
made
a
quite
intensive
study
of
the
mathematical
content
of
recent
blog
posts
by
adherents
of
the
RCS.
Despite
all
of
these
efforts,
the
only
justification
for
the
logical
cornerstone
RCS-identification
of
(RC-Θ)
that
we
[i.e.,
I
myself,
together
with
the
many
mathematicians
with
whom
I
have
discussed
these
issues]
could
find
either
in
oral
explanations
during
the
discussions
of
March
2018
or
in
subsequent
written
records
produced
by
adherents
of
the
RCS
[i.e.,
such
as
the
10pp.
manuscripts
referred
to
above
or
various
blog
posts]
were
statements
of
the
form
“I
don’t
see
why
not”.
[I
continue
to
find
it
utterly
bizarre
that
such
justifications
of
the
assertions
of
the
RCS
appear
to
be
taken
seriously
by
some
professional
mathematicians.]
In
particular,
we
were
unable
to
find
any
detailed
mathematical
discussion
by
adherents
of
the
RCS
of
the
fundamental
“(RC-Θ)
vs.
∧-property”
conflict
mentioned
above.
That
is
to
say,
in
summary,
the
mathematical
justification
for
the
“redundancy”
asserted
in
the
logical
cornerstone
assertion
(RC-Θ)
of
the
RCS
remains
a
complete
mystery
to
myself,
as
well
as
to
all
of
the
mathematicians
that
I
have
consulted
concerning
this
issue
[cf.
the
discussion
of
Examples
2.4.5,
2.4.7,
2.4.8].
Put
another
way,
the
response
of
all
of
the
mathematicians
with
whom
I
have
had
technically
meaningful
discussions
concerning
the
assertions
of
the
RCS
was
completely
uniform
and
unanimous,
i.e.,
to
the
effect
that
these
assertions
of
the
RCS
were
obviously
completely
math-
ematically
inaccurate/absurd,
and
that
they
had
no
idea
why
adherents
of
the
RCS
continued
to
make
such
manifestly
absurd
assertions.
In
particular,
it
should
be
emphasized
that
I
continue
to
search
for
a
professional
mathematician
[say,
in
the
field
of
arithmetic
geometry]
who
feels
that
he/she
understands
the
mathematical
content
of
the
assertions
of
the
RCS
and
is
willing
to
discuss
this
math-
ematical
content
with
me
or
other
mathematicians
with
whom
I
am
in
contact
[cf.
the
text
at
the
beginning
of
[Dsc2018]].
It
is
worth
noting
that
this
situation
also
constitutes
a
serious
violation
of
article
(6.)
Mathematicians
should
not
make
public
claims
of
potential
new
theorems
or
the
resolution
of
particular
mathematical
problems
unless
they
are
able
to
provide
full
details
in
a
timely
manner.
of
the
subsection
entitled
“Responsibilities
of
authors”
of
the
Code
of
Practice
of
the
European
Mathematical
Society
(cf.
[EMSCOP]).
In
this
context,
one
important
observation
that
should
be
kept
in
mind
is
the
following
[cf.
the
discussion
of
[Rpt2018],
§18]:
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
11
(UndIg)
There
is
a
fundamental
difference
between
(UndIg1)
criticism
of
a
mathematical
theory
that
is
based
on
a
solid,
technically
accurate
understanding
of
the
content
and
logi-
cal
structure
of
the
theory
and
(UndIg2)
criticism
of
a
mathematical
theory
that
is
based
on
a
funda-
mental
ignorance
of
the
content
and
logical
structure
of
the
theory.
An
elementary
classical
example
of
this
sort
of
difference
is
discussed
in
§2.1
below.
In
the
case
of
the
RCS,
the
lack
of
any
thorough
mathematical
discussion
of
the
fundamental
“(RC-Θ)
vs.
∧-property”
conflict
mentioned
above
in
the
vari-
ous
oral/written
explanations
set
forth
by
adherents
of
the
RCS
demonstrates,
in
a
definitive
way,
that
none
of
the
adherents
of
the
RCS
has
a
solid,
technically
accu-
rate
understanding
of
the
logical
structure
of
inter-universal
Teichmüller
theory
in
its
original
form,
i.e.,
in
particular,
of
the
central
role
played
in
this
logical
structure
by
the
“∧-property”
of
the
Θ-link.
Put
another
way,
the
only
logically
consistent
explanation
of
this
state
of
affairs
is
that
the
theory
“RCS-IUT”
that
adherents
of
the
RCS
have
in
mind,
i.e.,
the
theory
that
is
the
object
of
their
criticism,
is
simply
a
completely
different
—
and
logically
unrelated
—
theory
from
the
theory
constituted
by
inter-universal
Teichmüller
theory
in
its
original
form.
Finally,
it
should
be
mentioned
that
although
some
people
have
asserted
par-
allels
between
the
assertions
of
the
RCS
and
the
fundamental
error
in
the
first
version
of
Wiles’s
proof
of
the
Modularity
Conjecture
in
the
mid-1990’s,
this
anal-
ogy
is
entirely
inappropriate
for
numerous
reasons.
Indeed,
as
is
well-known,
nothing
even
remotely
close
to
the
phenomena
discussed
thus
far
in
the
present
§1.3
occurred
in
the
case
of
the
error
in
the
first
version
of
Wiles’s
proof.
The
fact
that
there
was
indeed
a
fatal
error
in
the
first
version
of
Wiles’s
proof
was
never
disputed
in
any
way
by
any
of
the
parties
involved;
the
only
issue
that
arose
was
the
issue
of
whether
or
not
the
proof
could
be
fixed.
By
contrast,
no
essential
er-
rors
have
been
found
in
inter-universal
Teichmüller
theory,
since
the
four
preprints
[IUTchI-IV]
on
inter-universal
Teichmüller
theory
were
released
in
August
2012.
That
is
to
say,
in
a
word,
the
assertions
of
the
RCS
are
nothing
more
than
mean-
ingless,
superficial
misunderstandings
of
inter-universal
Teichmüller
theory
on
the
part
of
people
who
are
clearly
not
operating
on
the
basis
of
a
solid,
technically
accurate
understanding
of
the
mathematical
content
and
essential
logical
structure
of
inter-universal
Teichmüller
theory.
§1.4.
The
importance
of
extensive,
long-term
interaction
In
general,
the
transmission
of
mathematical
ideas
between
individuals
who
share
a
sufficient
stock
of
common
mathematical
culture
may
be
achieved
in
a
relatively
efficient
way
and
in
a
relatively
brief
amount
of
time.
Typical
ex-
amples
of
this
sort
of
situation
in
the
context
of
interaction
between
professional
mathematicians
include
·
one-hour
mathematical
lectures,
·
week-long
mathematical
lecture
series,
and
·
informal
mathematical
discussions
for
several
days
to
a
week.
12
SHINICHI
MOCHIZUKI
In
the
context
of
mathematical
education,
typical
examples
include
·
written
or
oral
mathematical
examinations
and
·
mathematics
competitions.
The
successful
operation
of
each
of
these
examples
relies,
in
an
essential
way,
on
a
common
framework
of
mathematical
culture
that
is
shared
by
the
various
participants
in
the
activity
under
consideration.
On
the
other
hand,
in
the
case
of
a
fundamentally
new
area
of
research,
such
as
inter-universal
Teichmüller
theory,
which
evolved
out
of
research
over
the
past
quarter
of
a
century
concerning
absolute
anabelian
geometry,
certain
types
of
categories
arising
from
arithmetic
geometry,
and
certain
arithmetic
aspects
of
theta
functions,
the
collection
of
mathematicians
who
share
such
a
sufficient
stock
of
common
mathematical
culture
tends
to
be
relatively
small
in
number.
In
par-
ticular,
for
most
mathematicians
—
even
many
arithmetic
geometers
or
anabelian
geometers
—
short-term
interaction
of
the
sort
that
occurs
in
the
various
typical
examples
mentioned
above
is
far
from
sufficient
to
achieve
an
effective
trans-
mission
of
mathematical
ideas.
That
is
to
say,
no
matter
how
mathematically
talented
the
participants
in
such
platforms
of
interaction
may
be,
it
takes
time
for
the
participants
to
·
analyze
and
sort
out
numerous
mutual
misunderstandings,
·
develop
effective
techniques
of
communication
that
can
transcend
such
misunderstandings,
and
·
digest
and
absorb
new
ideas
and
modes
of
thought.
Depending
on
the
mathematical
content
under
consideration,
as
well
as
on
the
mathematical
talent,
mathematical
background,
and
time
constraints
of
the
partic-
ipants,
this
painstaking
process
of
analysis/development/digestion/absorption
may
require
patiently
sustained
efforts
to
continue
constructive,
orderly
mathematical
discussions
[via
e-mail,
online
video
discussions,
or
face-to-face
meetings]
over
a
period
of
months
or
even
years
to
reach
fruition.
Indeed,
my
experience
in
exposing
the
ideas
of
inter-universal
Teichmüller
theory
to
numerous
mathematicians
over
the
past
decade
suggests
strongly
that,
in
the
case
of
inter-universal
Teichmüller
theory,
it
is
difficult
to
expedite
this
process
to
the
extent
that
it
can
be
satisfactorily
achieved
in
less
than
half
a
year
or
so.
In
particular,
in
the
case
of
inter-universal
Teichmüller
theory,
a
week-long
session
of
discussions
such
as
the
discussions
held
at
RIMS
in
March
2018
with
two
adherents
of
the
RCS
[cf.
[Rpt2018],
[Dsc2018]]
is
far
from
sufficient.
This
is
something
that
I
emphasized,
both
orally
during
these
discussions
and
in
e-mails
to
these
two
adherents
of
the
RCS
during
the
summer
of
2018
subsequent
to
these
discussions.
§1.5.
The
historical
significance
of
detailed,
explicit,
accessible
records
As
was
discussed
in
§1.3,
I
continue
to
search
for
a
professional
mathematician
[say,
in
the
field
of
arithmetic
geometry]
who
purports
to
understand
the
math-
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
13
ematical
justification
for
the
RCS-redundancy
asserted
in
the
logical
corner-
stone
assertion
(RC-Θ)
—
i.e.,
in
particular,
who
has
confronted
the
mathematical
content
of
the
fundamental
“(RC-Θ)
vs.
∧-property”
conflict
mentioned
in
§1.3
—
and
who
is
prepared
to
discuss
this
mathematical
content
with
me
or
other
mathematicians
with
whom
I
am
in
contact.
Of
course,
a
detailed,
explicit,
mathematically
substantive,
and
readily
ac-
cessible
written
exposition
—
i.e.,
as
an
alternative
to
direct
mathematical
discussions
[via
e-mail,
online
video
discussions,
or
face-to-face
meetings]
—
of
the
mathematical
justification
for
the
log-
ical
cornerstone
assertion
(RC-Θ)
would
also
be
quite
welcome
[cf.
the
discussion
of
[Rpt2014],
(7)].
Moreover,
in
this
context,
it
should
be
emphasized
that
such
a
detailed,
explicit,
mathematically
substantive,
and
readily
accessible
written
expo-
sition
would
be
of
great
value
not
only
for
professional
mathematicians
and
graduate
students
who
are
involved
with
inter-universal
Teichmüller
theory
at
the
present
time,
but
also
for
scholars
in
the
[perhaps
distant!]
future.
In
general,
it
cannot
be
overemphasized
that
maintaining
such
detailed,
ex-
plicit,
mathematically
substantive,
and
readily
accessible
written
records
is
of
fundamental
importance
to
the
development
of
mathematics.
Indeed,
as
was
discussed
in
the
final
portion
of
[Rpt2018],
§3,
from
a
historical
point
of
view,
it
is
only
by
maintaining
such
written
records
that
the
field
of
math-
ematics
can
avoid
the
sort
of
well-known
and
well-documented
confusion
that
lasted
for
so
many
centuries
concerning
“Fermat’s
Last
Theorem”.
Moreover,
it
is
fascinating
to
re-examine,
from
the
point
of
view
of
a
modern
observer,
the
in-
tense
debates
that
occurred,
during
the
time
of
Galileo,
concerning
the
theory
of
heliocentrism
or,
during
the
time
of
Einstein,
concerning
the
theory
of
relativity.
Perhaps
a
more
pertinent
example
[as
was
pointed
out
to
the
author
by
Fu-
miharu
Kato],
relative
to
the
issue
of
“redundant
copies”,
may
be
found
in
the
well-known
dispute
—
i.e.,
concerning
“algebraic
truths”
[which
corresponds
to
Weierstrass’
approach]
versus
“geometric
fantasies”
[which
corresponds
to
Rie-
mann’s
approach]
—
in
the
19-th
century
between
Weierstrass
and
Riemann
con-
cerning
approaches
to
complex
function
theory
[cf.
[Btt]].
That
is
to
say,
the
criticism
by
Weierstrass
of
the
“geometric
fantasies”
approach
due
to
Riemann
via
analytic
continuation
on
Riemann
surfaces,
which
are
obtained
by
gluing
to-
gether
—
what
are,
perhaps
to
some
observers,
seemingly
“redundant”!
—
copies
of
open
subsets
of
the
complex
plane
[cf.
the
discussion
of
the
well-known
gluing
construction
of
the
projective
line
in
Example
2.4.7
below;
the
discussion
of
ana-
lytic
continuation
of
the
complex
logarithm
in
the
discussion
surrounding
(FxEuc)
in
§3.1
below,
as
well
as
in
Example
3.10.1
below;
the
illustrations
of
[AnCnCv],
[AnCnLg]!],
exhibits
remarkable
parallels
in
spirit
to
the
assertions
of
the
RCS.
Here,
it
is
also
of
interest
to
note
[as
was
pointed
out
to
the
author
by
Ben-
jamin
Collas]
that
the
issue
of
analytic
continuation
of
the
complex
logarithm
is
very
closely
related
to
the
long
and
heated
dispute
between
Leibniz
and
[Johann]
Bernoulli
concerning
the
appropriate
definition
of
logarithms
of
negative
and
more
general
complex
numbers
—
a
dispute
that
was
ultimately,
in
some
sense,
resolved
by
Euler’s
formula
and
the
acceptance
of
the
multi-valued
nature
of
the
complex
14
SHINICHI
MOCHIZUKI
logarithm
[cf.
the
discussion
of
“Euler’s
Formula
and
Its
Consequences”
in
[AnHst],
Chapter
I,
§I.5,
although
it
must
be
kept
in
mind
that
in
Euler’s
time,
complex
function
theory,
i.e.,
of
the
sort
necessary
to
treat
the
complex
logarithm
as
a
holomorphic
function,
had
not
yet
been
developed].
This
multi-valued
nature
of
the
complex
logarithm
is
closely
related
not
only
to
the
theory
of
analytic
continuation
of
the
complex
logarithm,
but
also
to
the
theory
of
coverings
[in
the
sense
of
the
classical
topological
fundamental
group],
where
the
indeterminacy
in
values
may
be
understood
as
an
indeterminacy
in
the
choice
of
basepoint,
that
is
to
say,
as
a
sort
of
distant
—
though
nonetheless
quite
direct!
—
ancestor
of
the
inter-universal
indeterminacies
that
appear
in
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
§3.8,
especially
Example
3.8.1,
below!].
The
central
role
occupied
by
the
notion
of
analytic
continuation
in
these
his-
torical
disputes
of
Weierstrass/Riemann
and
Leibniz/Bernoulli
is
also
of
interest,
in
the
context
of
inter-universal
Teichmüller
theory,
in
light
of
the
central
role
played,
in
the
history
of
analytic
continuation,
by
the
functional
equation
[i.e.,
Jacobi
identity]
for
the
theta
function,
which
exhibits
numerous
remarkable
structural
similarities
to
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
Example
3.3.2
below],
and
which
is
also
closely
related
to
Riemann’s
famous
research
con-
cerning
the
theory
of
the
functional
equation/analytic
continuation
of
the
Riemann
zeta
function.
Research
in
the
19-th
century
concerning
analytic
continuation
may
also
be
regarded
as
a
sort
of
early
precursor
of
more
modern
notions
such
as
monodromy
representations
and
connections/crystals
—
an
observation
which
is
of
interest
in
light
of
the
analogy
between
connections/crystals
and
the
notion
of
multiradiality
in
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
[Alien],
§3.1,
(v),
as
well
as
the
discussion
of
§3.5,
§3.10
below].
Before
proceeding,
we
remark
that
it
is
interesting
to
observe
that
this
his-
torical
discussion
of
the
functional
equation
of
the
theta
function
in
the
context
of
analytic
continuation,
in
this
case
on
the
upper
half-plane,
is
reminiscent
of
the
famous
observation
of
Poincaré
—
i.e.,
in
the
form
of
an
idea
that
came
to
him
while
in
transit
during
his
travels
—
that
“the
transformations”
that
he
had
used
“to
define
Fuchsian
functions
were
identical
with
those
of
non-Euclidian
geometry”
[cf.
[Pnc],
p.
53].
This
famous
observation
concerning
the
isomorphic
nature
of
the
group
of
transformations
of
a
modular
function
—
i.e.,
such
as
the
theta
function
—
and
a
certain
group
of
symmetries
of
the
[“non-Euclidean”]
hyperbolic
geometry
of
the
upper
half-plane
seems
to
be
one
of
the
principal
motivations
behind
the
famous
quote,
due
to
Poincaré,
that
“mathematics
is
the
art
of
giving
the
same
name
to
different
things”,
i.e.,
“things
which
differ
in
matter,
but
are
similar
in
form”
[cf.
the
discussion
of
[Pnc],
pp.
34
–
35].
Moreover,
this
remarkable
train
of
thought,
which
was
recorded
for
posterity
by
Poincaré
in
[Pnc],
is
of
particular
interest
in
the
context
of
the
present
discussion
of
the
relationship
between
various
classical
notions
and
inter-universal
Teichmüller
theory
in
that
it
seems
almost
prescient
in
its
deep
resemblance
to
the
main
notions
—
namely,
coricity
and
multiradiality
—
that
underlie
the
concept
of
inter-universality
in
inter-universal
Teichmüller
theory.
Indeed:
(SmDff1)
The
search
for
coric
structures
in
inter-universal
Teichmüller
theory
may
be
thought
of
precisely
as
the
search
for
the
“same
name”
for
suitable
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
15
portions
of
structures
on
opposite
sides
—
i.e.,
“different
things”
—
of
the
Θ-
or
log-links
in
inter-universal
Teichmüller
theory.
Moreover,
perhaps
the
most
fundamental
instance
of
such
a
coric
structure
in
inter-universal
Teichmüller
theory
consists
of
the
abstract
topological
groups,
regarded
up
to
isomorphism,
underlying
the
Galois
groups/arithmetic
fundamental
groups
that
arise
from
rings/schemes
on
opposite
sides
of
the
Θ-
or
log-
links.
(SmDff2)
The
search
for
multiradial
structures
in
inter-universal
Teichmüller
the-
ory
may
be
thought
of
precisely
as
the
search
for
the
“same
name”
—
i.e.,
more
precisely,
in
the
form
of
an
isomorphism
up
to
suitable
indetermina-
cies
—
for
·
a
suitable
portion
of
the
system
[or
portion
of
the
log-theta-
lattice]
consisting
of
distinct
ring/scheme
structures
linked
by
the
Θ-link,
on
the
one
hand,
and
·
a
suitable
portion
of
a
single
ring/scheme
structure,
on
the
other
—
i.e.,
for
“things
which
differ
in
matter”.
More
technical
details
may
be
found
in
the
discussion
of
coricity
and
inter-universality
in
§3.2,
§3.8,
below,
as
well
in
the
discussion
of
descent
via
the
multiradial
algorithms
of
inter-universal
Teichmüller
theory
given
in
§3.10,
§3.11,
below.
Of
course,
from
the
point
of
view
of
a
modern
observer
who
is
well-versed
in
axiomatic
set
theory
and
the
theory
of
[Grothendieck]
universes,
the
bitter
historical
disputes
of
Weierstrass/Riemann
and
Leibniz/Bernoulli
discussed
above
seem
somewhat
quaint
or
even
“silly”.
One
should
not,
however,
in
this
context,
overlook
the
importance
of
such
bitter
historical
disputes
in
motivating
the
develop-
ment
of
such
modern
tools
as
axiomatic
set
theory
and
the
theory
of
[Grothendieck]
universes,
which
underlie
this
privileged
viewpoint
of
the
modern
observer.
Con-
versely,
when
viewed
from
this
historical
perspective,
the
assertions
of
the
RCS
seem
all
the
more
like
a
sort
of
bizarre
anachronism,
which
has
no
place
in
the
21-st
century
[cf.,
especially,
the
entirely
elementary
nature
of
the
various
examples
—
such
as
the
gluing
construction
of
the
projective
line
discussed
in
Example
2.4.7
—
that
appear
in
§2
below]!
Moreover,
in
the
case
of
the
Leibniz/Bernoulli
dispute,
it
is
of
interest
to
note
—
especially,
in
the
context
of
the
highly
sensationalist
nature
of
the
coverage
of
inter-universal
Teichmüller
theory
in
the
English-language
mass-media
and
internet
[cf.,
e.g.,
the
discussion
of
§1.8,
§1.10,
§1.12
below]
—
that
apparently,
at
the
time
of
the
dispute,
the
dispute
was
kept
as
secret
as
possible
in
order
to
avoid
“damaging
the
prestige
of
pure
mathematics
as
an
exact
and
rigorous
science”
[cf.
the
discussion
of
“Euler’s
Formula
and
Its
Consequences”
in
[AnHst],
Chapter
I,
§I.5].
At
any
rate,
in
the
context
of
this
sort
of
historical
discussion,
it
cannot
be
overemphasized
that
all
of
these
historical
re-examinations
[i.e.,
of
the
sort
that
underlie
the
discussion
of
the
last
few
paragraphs!]
are
technically
possible
precisely
because
of
the
existence
of
detailed,
explicit,
mathematically
sub-
stantive,
and
readily
accessible
written
expositions
of
the
logical
structure
underlying
the
various
central
assertions
that
arose
in
the
de-
bate.
16
SHINICHI
MOCHIZUKI
In
this
context,
it
should
be
noted
that,
from
a
historical
point
of
view,
one
pattern
that
typically
underlies
the
formidable
deadlocks
that
tend
to
occur
in
such
debates
is
the
point
of
view,
on
the
part
of
parties
opposed
to
a
newly
developed
theory,
that
(CmSn)
it
is
a
“matter
of
course”
or
“common
sense”
—
i.e.,
in
the
lan-
guage
of
the
above
discussion,
a
matter
that
is
so
profoundly
self-evident
that
any
“decent,
reasonable
observer”
would
undoubtedly
find
detailed,
explicit,
mathematically
substantive,
and
readily
accessible
written
expo-
sitions
of
its
logical
structure
to
be
entirely
unnecessary
—
that
the
issues
under
consideration
can
be
completely
resolved
within
some
exist-
ing,
familiar
framework
of
thought
without
the
introduction
of
the
newly
developed
theory,
which
is
regarded
as
deeply
disturbing
and
unlikely
to
be
of
use
in
any
substantive
mathematical
sense
—
cf.,
e.g.,
the
discussion
of
Example
1.5.1
below.
In
fact,
however,
(OvDlk)
ultimately,
the
only
meaningful
technical
tool
that
humanity
can
apply
to
develop
the
cultural
infrastructure
necessary
to
overcome
such
deadlocks
is
precisely
the
production
of
detailed,
explicit,
mathematically
substantive,
and
readily
accessible
written
expositions
of
the
logical
structure
underlying
the
points
of
view
that
are
regarded
by
certain
parties
as
being
a
“matter
of
common
sense”
[cf.
the
discussion
of
[EMSCOP]
in
§1.3;
the
discussion
of
(UndIg)
in
§1.3;
the
discussion
of
§1.10
below;
§2.1
below;
the
discussion
of
(FxRng),
(FxEuc),
(FxFld),
(RdVar)
in
§3.1
below].
Example
1.5.1:
Irrationality,
impartiality,
and
the
Voodoo
Hypothesis.
(i)
We
begin
by
recalling
the
very
classical
and
elementary
proof
that,
for
√
any
prime
number
p,
p
is
irrational,
i.e.,
that
there
does
not
exist
any
rational
number
whose
square
is
equal
to
p.
Indeed,
if
there
did
exist
a
rational
number
r
=
a/b,
where
a
and
b
are
relatively
prime
nonzero
integers,
such
that
r
2
=
p,
then
the
resulting
relation
a
2
=
p
·
b
2
would
violate
the
unique
factorization
property
satisfied
by
the
nonzero
integers.
This
unique
factorization
property
may,
in
turn,
by
verified
by
applying
the
Euclidean
division
algorithm.
(ii)
The
discovery
of
the
irrationality
of
square
roots
of
prime
numbers
dis-
cussed
in
(i)
is
typically
attributed,
in
the
case
p
=
2
[in
which
case
the
Euclidean
division
algorithm
amounts,
in
essence,
to
the
classification
of
integers
into
odd
and
even
integers],
to
the
ancient
Greek
philosopher
Hippasus.
Apparently,
this
discovery
arose
in
the
context
of
applying
the
Pythagorean
Theorem
[cf.
the
discus-
sion
of
Examples
1.10.1,
1.12.1
below]
to
compute
the
length
of
the
diagonal
of
a
square
with
sides
of
length
1.
The
Pythagorean
school
was
reported
to
have
found
this
discovery
to
be
shocking
and
deeply
disturbing,
and
indeed
it
seems
that
this
negative
appraisal
of
Hippasus’
discovery
may
have
led
to
the
death
of
Hippasus
by
drowning.
(iii)
The
sort
of
negative
appraisal
that
occurred
in
(ii)
could
easily
arise
in
the
mind
of
any
observer
—
indeed,
even
modern
observers
[such
as
students]
who
are
not
familiar
with
the
sort
of
abstract
mathematical
reasoning
that
underlies
the
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
17
proof
of
(i)
—
who
has
a
deep
sense
of
confidence
in
his/her
understanding
of
the
“common
sense
definition
of
mathematics”
as
the
study
of
explicit
computations
involving
rational
numbers,
e.g.,
of
the
sort
that
may
be
done
with
a
desktop
calculator
[or,
in
earlier
centuries,
with
an
abacus!].
[We
refer
to
Example
1.5.2
below
for
further
similar
examples
of
this
sort
of
phenomenon.]
Such
a
negative
appraisal
on
the
part
of
a
strongly
computationally
oriented
observer
might
also
be
additionally
supported
by
(HeurBlf)
a
heuristically
supported
belief
that
“surely”
if
one
is
given
sufficient
time
and
manpower
to
perform
suitable
computations
to
sufficiently
high
order,
then
there
is
no
doubt
that
one
should
be
able
to
find
a
rational
square
root
of
2.
Moreover,
depending
on
the
social/political
circumstances
surrounding
the
situa-
tion,
even
third
parties
who
are
ignorant
of
the
details
of
situation
might
be
led
(FsObjImpl)
to
[falsely!]
assert
that
the
only
truly
objective
or
impartial
way
√
to
treat
the
two
schools
—
i.e.,
of
people
who
assert
the
irrationality
of
2
via
the
proof
of
(i)
and
people
who
√
doubt
this
argument
—
is
to
talk
as
if
the
issue
of
the
irrationality
of
2
is
unresolvable
or
even
to
refuse
to
discuss
the
issue
at
all.
Indeed,
it
is
worth
recalling
that
ancient
accounts
of
Hippasus
and
the
Pythagorean
school
are
a
stark
reminder
of
just
how
dire
the
consequences
can
be
when
those
who
are
socially/politically
recognized
as
“objective
arbiters”
for
a
mathematical
dispute
act
on
the
basis
of
a
grossly
mathematically
inaccurate
understanding
of
the
situation.
(iv)
In
the
context
of
the
discussion
of
(iii),
it
is
useful
to
refer
to
a
situation
[i.e.,
such
as
the
situation
discussed
in
(iii)]
in
which
the
validity
of
a
mathematical
proof
is
called
into
question,
not
on
the
basis
of
some
sort
of
logical
defect
[i.e.,
such
as
a
gap
in
the
proof],
but
on
the
basis
of
some
sort
of
heuristically
based
belief
[cf.
(HeurBlf)]
to
the
effect
that
“surely”
if
one
is
given
sufficient
time
and
manpower
to
sort
through
various
technical
details,
then
there
is
no
doubt
that
one
should
be
able
to
find
some
sort
of
substantive
problem
with
the
proof
—
such
as
a
counterexample
or
fallacious
reasoning,
i.e.,
some
sort
of
“voodoo”
—
as
an
invocation
of
the
“Voodoo
Hypothesis”.
Once
the
Voodoo
Hypothesis
has
been
invoked,
mathematicians
who
have
a
mathematically
accurate,
rigorous
understanding
of
the
proof
in
question
are
then
often
portrayed
as
being
nothing
more
than
mindlessly
obedient
zombies,
i.e.,
who
are
acting
solely
or
essentially
under
the
influence
of
the
“voodoo”
applied
in
the
proof.
Indeed,
this
is
precisely
the
sort
of
situation
that
has
developed,
in
certain
sectors
of
the
mathematical
community,
concerning
inter-universal
Teichmüller
theory
—
cf.
the
discussion
of
RCS-IUT
in
§1.2,
§1.3,
as
well
as
§1.8,
§1.10
below;
Examples
2.4.5,
2.4.7,
2.4.8
below;
the
discussion
of
(FxRng),
(FxEuc),
(FxFld),
(RdVar)
in
§3.1
below.
(v)
At
this
point,
we
observe
that
the
discussion
of
(ii),
(iii),
and
(iv)
prompt
the
following
question:
What
lessons
may
be
learned
from
the
discussion
in
(ii),
√
(iii),
and
(iv)
of
the
negative
appraisal
of
the
proof
of
the
irrationality
of
2?
18
SHINICHI
MOCHIZUKI
First
of
all,
it
is
important
to
remember
that
(Blf
=Pf)
heuristically
based
beliefs,
as
in
(HeurBlf),
are
not
mathematical
proofs
and,
in
particular,
can
never
serve
as
a
“viable
substitute”
for
a
rigorous
mathematical
proof.
Secondly,
it
is
important
to
remember
[cf.
(FsObjImpl)]
that
(MthAcc)
a
truly
objective/impartial
position
concerning
a
mathematical
dis-
pute
can
only
be
achieved
as
a
result
of
a
mathematically
accurate
understanding
of
the
mathematics
involved
and,
in
particular,
can
never
be
achieved
through
an
ignorance
of
the
mathematics
involved.
In
this
context,
it
is
interesting
to
note
that
(MthAcc)
has
important
consequences
from
the
point
of
view
of
the
topic
of
computer
verification
of
mathematical
assertions
[cf.
the
discussion
surrounding
(CmpVer)
in
§1.12
below].
That
is
to
say,
for
instance,
in
the
case
of
the
proof
of
(i),
although
it
is
not
so
technically
difficult
to
formalize
this
[relatively
simple!]
argument
in
such
a
way
that
the
argument
could
be
“verified”
by
a
computer,
(SocPol)
such
a
computer
verification
becomes
entirely
socially/politically
meaningless
in
situations
in
which
parties
—
such
as
the
computation-
ally
oriented
observer
of
(iii)!
—
who
do
not
share
or
recognize
the
abstract
conceptual
mathematical
framework
that
necessarily
underlies
any
sort
of
computer-ready
formalization
of
the
proof
of
(i)
hold
a
monopoly
on
so-
cial/political
authority.
Thus,
in
summary,
from
a
historical
point
of
view,
it
seems
that
the
main
lesson
to
be
learned
from
the
situation
discussed
in
(ii),
(iii),
and
(iv)
is
(LTInfr)
the
fundamental
importance
—
in
the
context
of
dealing
in
a
mean-
ingful
and
effective
manner
with
mathematical
disputes
—
of
maintain-
ing
a
long-term
infrastructural
apparatus
for
directly
confronting,
disseminating,
and
further
developing
the
mathematics
involved
[cf.
the
discussion
surrounding
(BlkAcc)
in
§1.12
below].
Needless
to
say,
the
starting
point
of
the
activities
of
such
a
long-term
infras-
tructural
apparatus
necessarily
lies
in
the
production
of
detailed,
explicit,
mathe-
matically
substantive,
and
readily
accessible
written
expositions
of
the
logical
structure
underlying
the
points
of
view
that
are
regarded
by
various
parties
as
being
a
“matter
of
common
sense”.
Example
1.5.2:
The
internet/mass
media
as
an
apple
of
discord.
In
Ex-
ample
1.5.1
[cf.,
especially,
the
discussion
of
“modern
observers”
√
in
Example
1.5.1,
(iii)],
the
particular
example
of
the
issue
of
the
irrationality
of
2
was
examined
in
detail.
In
fact,
however,
this
sort
of
situation
exists
in
quite
substantial
abundance
throughout
mathematics
and
especially
throughout
the
sort
of
elementary
mathe-
matics
that
is
commonly
covered
in
primary
and
secondary,
as
well
as
in
university,
eduation.
Well-known
examples
of
this
phenomenon
include
the
following:
(NegInt)
the
sense,
in
the
context
of
multiplication
of
positive
and
negative
inte-
gers,
that
any
product
of
two
negative
integers
must
be
a
negative
integer,
i.e.,
on
the
grounds
that
the
appearance
of
“two
minus
signs”
in
the
prod-
uct
“surely”
results
in
output
that
is
“all
the
more
negative!”;
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
19
(MxDiag)
the
sense,
in
the
context
of
elementary
linear
algebra,
that
“surely”
all
matrices
are
diagonalizable,
i.e.,
if
one
just
tries
hard
enough
to
find
the
“right
basis!”.
Moreover,
unlike
the
case
with
the
ancient
context
discussed
in
Example
1.5.1,
(ii),
this
sort
of
phenomenon
can
be
substantially
further
exacerbated
in
modern
con-
texts
by
the
internet/mass
media,
which
exhibits
a
conspicuous
tendency
to
operate
as
a
sort
of
“apple
of
discord”
that
has
the
effect
of
not
only
creating,
but
also
amplifying
to
often
absurd
proportions,
an
artificial
socio-political
dynamic
that
fuels
a
burning
desire
to
leap
to
conclusions
and
achieve
“instant
satisfaction”
with
respect
to
some
sort
of
essentially
meaningless
fictional/delusional
mirage,
e.g.,
with
respect
to
achieving
an
understanding
of
some
sort
of
apparently
puzzling
mathe-
matical
phenomenon
of
the
sort
described
in
(NegInt),
(MxDiag)
that
appears,
to
some
observers,
to
defy
their
deep
heuristic
“common
sense”
understanding
of
the
situation
—
cf.
the
discussion
of
§1.8,
§1.10,
§1.12,
below.
Finally,
we
observe
that,
just
as
in
the
case
of
the
discussion
surrounding
(LTInfr)
in
Example
1.5.1,
(v),
the
only
way,
to
the
knowledge
of
the
author
at
the
time
of
writing,
to
overcome
the
detrimental
effects
of
such
artificial
socio-political
dynamics
lies
in
maintaining
a
long-term
infrastructural
apparatus
for
directly
confronting,
disseminating,
and
further
developing
the
mathematics
involved,
the
starting
point
of
which
inevitably
involves
the
production
of
detailed,
explicit,
mathematically
substantive,
and
readily
accessible
written
expositions
of
the
logical
structure
underlying
the
points
of
view
that
are
regarded
by
various
parties
as
being
a
“matter
of
common
sense”.
§1.6.
The
importance
of
further
dissemination
One
fundamental
and
frequently
discussed
theme
in
the
further
development
of
inter-universal
Teichmüller
theory
is
the
issue
of
increasing
the
number
of
profes-
sional
mathematicians
who
have
a
solid,
technically
accurate
understanding
of
the
details
of
inter-universal
Teichmüller
theory.
Indeed,
this
issue
is
in
some
sense
the
central
topic
of
[Rpt2013],
[Rpt2014].
As
discussed
in
§1.4,
in
order
to
achieve
such
a
solid,
technically
accurate
understanding
of
the
theory,
it
is
neces-
sary
to
devote
a
substantial
amount
of
time
and
effort
over
a
period
of
roughly
half
a
year
to
two
or
three
years,
depending
on
various
factors.
It
also
requires
the
participation
of
professional
mathematicians
or
graduate
students
who
are
·
sufficiently
familiar
with
numerous
more
classical
theories
in
arithmetic
geometry
[cf.
the
discussion
of
[Alien],
§4.1,
(ii);
[Alien],
§4.4,
(ii)],
·
sufficiently
well
motivated
and
enthusiastic
about
studying
inter-universal
Teichmüller
theory,
and
·
sufficiently
mathematically
talented,
and
who
have
a
·
sufficient
amount
of
time
to
devote
to
studying
the
theory.
As
a
result
of
quite
substantial
dissemination
efforts
not
only
on
my
part,
but
also
on
the
part
of
many
other
mathematicians,
the
number
of
professional
mathe-
maticians
who
have
achieved
a
sufficiently
detailed
understanding
of
inter-universal
Teichmüller
theory
to
make
independent,
well
informed,
definitive
statements
con-
cerning
the
theory
that
may
be
confirmed
by
existing
experts
on
the
theory
[cf.
also
the
discussion
of
§1.7
below]
is
roughly
on
the
order
of
10.
It
is
worth
noting
that
20
SHINICHI
MOCHIZUKI
although
this
collection
of
mathematicians
is
centered
around
RIMS,
Kyoto
Uni-
versity,
it
includes
mathematicians
of
many
nationalities
and
of
age
ranging
from
around
30
to
around
60.
One
recent
example
demonstrated
quite
dramatically
that
it
is
quite
possible
to
achieve
a
solid
mathematical
understanding
of
inter-universal
Teichmüller
theory
as
a
graduate
student
by
studying
on
one’s
own,
outside
of
Japan,
and
with
essentially
zero
contact
with
RIMS,
except
for
a
very
brief
pe-
riod
of
a
few
months
at
the
final
stage
of
the
student’s
study
of
inter-universal
Teichmüller
theory.
Finally,
we
observe,
in
the
context
of
the
discussion
[cf.
§1.3,
§1.4,
§1.5]
of
the
assertions
of
the
RCS,
that
another
point
that
should
be
emphasized
is
that
it
is
also
of
fundamental
importance
to
increase
the
number
of
professional
mathematicians
[say,
in
the
field
of
arithmetic
geometry]
who
have
a
solid
technical
understanding
of
the
mathematical
content
of
the
assertions
of
the
RCS,
and
who
are
pre-
pared
to
discuss
this
mathematical
content
with
members
of
the
“IUT
community”
[i.e.,
with
mathematicians
who
are
substantially
involved
in
mathematical
research
and/or
dissemination
activities
concerning
inter-universal
Teichmüller
theory].
Here,
we
note
in
passing
that
such
a
solid
technical
understanding
of
the
mathematical
content
of
the
assertions
of
the
RCS
is
by
no
means
“equivalent”
to
expressions
of
support
for
the
RCS
on
the
basis
of
non-mathematical
—
i.e.,
such
as
social,
political,
or
psychological
—
reasons.
In
this
context,
it
should
also
be
emphasized
and
understood
[cf.
the
discussion
of
[Rpt2014],
(7)]
that
both
·
producing
detailed,
explicit,
mathematically
substantive,
and
readily
ac-
cessible
written
expositions
of
the
mathematical
justification
of
assertions
of
the
RCS
[such
as
(RC-Θ)!],
i.e.,
as
discussed
in
§1.5,
and
·
increasing
the
number
of
professional
mathematicians
[say,
in
the
field
of
arithmetic
geometry]
who
have
a
solid
technical
understanding
of
the
mathematical
content
of
the
assertions
of
the
RCS,
and
who
are
prepared
to
discuss
this
mathematical
content
with
members
of
the
IUT
community
are
in
the
interest
not
only
of
the
IUT
community,
but
of
the
RCS
as
well.
More-
over,
the
process
of
attaining
a
solid,
technically
accurate
understanding
of
the
precise
logical
relationship
between
RCS-IUT
and
IUT,
i.e.,
as
ex-
posed,
for
instance,
in
the
present
paper,
can
serve
as
a
valuable
peda-
gogical
tool
[cf.
the
discussion
of
[Rpt2018],
§17]
for
mathematicians
currently
in
the
process
of
studying
inter-universal
Teichmüller
theory.
§1.7.
The
notion
of
an
“expert”
One
topic
that
sometimes
arises
in
the
context
of
discussions
of
dissemination
of
inter-universal
Teichmüller
theory
[i.e.,
as
in
§1.6],
is
the
following
issue:
What
is
the
definition
of,
or
criterion
for,
being
an
“expert”
on
inter-
universal
Teichmüller
theory?
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
21
In
a
word,
it
is
very
difficult
to
give
a
brief,
definitive
answer,
e.g.,
in
the
form
of
a
straightforward,
easily
applicable
criterion,
to
this
question.
On
the
other
hand,
in
this
context,
it
should
also
be
pointed
out
that
the
difficulties
that
arise
in
the
case
of
inter-universal
Teichmüller
theory
are,
in
fact,
not
so
qualitatively
different
from
the
difficulties
that
arise
in
answering
the
analogous
question
for
mathematical
theories
other
than
inter-universal
Teichmüller
theory.
These
difficulties
arise
throughout
the
daily
life
of
professional
mathematicians
in
numerous
contexts,
such
as
the
following:
(Ev1)
preparing
suitable
exercises
or
examination
problems
to
educate
and
evaluate
students,
(Ev2)
evaluating
junior
mathematicians,
(Ev3)
refereeing/evaluating
mathematical
papers
for
journals.
From
my
point
of
view,
as
the
author
of
[IUTchI-IV],
one
fundamental
criterion
that
I
always
keep
in
mind
—
not
only
the
in
case
of
[IUTchI-IV],
but
also
in
the
case
of
other
papers
that
I
have
written,
as
well
as
when
I
am
involved
in
the
various
types
of
evaluation
procedures
(Ev1)
∼
(Ev3)
discussed
above
—
is
the
issue
of
the
extent
to
which
the
level
of
understanding
of
the
mathematician
in
question
enables
the
mathematician
to
“stand
on
his/her
own
two
feet”
with
regard
to
various
assertions
concerning
the
theory,
on
the
basis
of
independent,
logical
reasoning,
without
needing
to
be
“propped
up”
or
cor-
rected
by
me
or
other
known
experts
in
the
theory.
I
often
refer
to
this
criterion
as
the
criterion
of
autonomy
of
understanding.
Of
course,
from
a
strictly
rigorous
point
of
view,
this
criterion
is,
in
some
sense,
not
so
“well-defined”
and,
in
many
contexts,
difficult
to
apply
in
a
straightforward
fashion.
On
the
other
hand,
in
the
past,
various
mathematicians
involved
with
inter-universal
Teichmüller
theory
have
demonstrated
such
an
autonomous
level
of
understanding
in
the
following
ways:
(Atm1)
the
ability
to
detect
various
minor
errors/oversights
in
[IUTchI-III];
(Atm2)
the
ability
to
propose
new,
insightful
ways
of
thinking
about
various
aspects
of
inter-universal
Teichmüller
theory;
(Atm3)
the
ability
to
propose
ways
of
modifying
inter-universal
Teichmüller
theory
so
as
to
yield
stronger
or
more
efficient
versions
of
the
theory;
(Atm4)
the
ability
to
produce
technically
accurate
oral
or
written
expositions
of
inter-universal
Teichmüller
theory;
(Atm5)
the
ability
to
supervise
or
direct
new
mathematicians
—
i.e.,
by
train-
ing/educating
professional
mathematicians
or
graduate
students
with
re-
gard
to
inter-universal
Teichmüller
theory
—
who,
in
due
time,
demon-
strate
various
of
the
four
types
of
ability
(Atm1)
∼
(Atm4)
discussed
above.
Of
course,
just
as
in
the
case
of
other
mathematical
theories,
different
experts
demonstrate
their
expertise
in
different
ways.
That
is
to
say,
experts
in
inter-
universal
Teichmüller
theory
often
demonstrate
their
expertise
with
respect
to
some
of
these
five
types
of
ability
(Atm1)
∼
(Atm5),
but
not
others.
22
SHINICHI
MOCHIZUKI
In
this
context,
it
should
be
pointed
out
that
one
aspect
of
inter-universal
Teichmüller
theory
that
is
currently
still
under
development
is
the
analogue
for
inter-universal
Teichmüller
theory
of
(Ev1),
i.e.,
preparing
suitable
exercises
for
mathematicians
currently
in
the
process
of
studying
inter-universal
Teichmüller
theory.
This
point
of
view
may
be
seen
in
the
discussion
in
the
final
portion
of
the
Introduction
to
[Alien],
as
well
as
in
the
discussion
of
“valuable
pedagogical
tools”
in
[Rpt2018],
§17.
Indeed,
many
of
the
technical
issues
discussed
in
[Rpt2018],
§15
[or,
alternatively,
Example
3.2.2
of
the
present
paper],
may
easily
be
reformu-
lated
as
“exercises”
or,
alternatively,
as
“examination
problems”
for
evaluating
the
level
of
understanding
of
mathematicians
in
the
process
of
studying
inter-universal
Teichmüller
theory.
§1.8.
Fabricated
versions
spawn
fabricated
dramas
As
discussed
in
§1.6,
§1.7,
by
now
there
is
a
substantial
number
of
mathe-
maticians
who
have
attained
a
thorough,
accurate,
and
automous
understanding
of
inter-universal
Teichmüller
theory.
In
each
of
the
cases
of
such
mathematicians
that
I
have
observed
thus
far,
such
an
understanding
of
the
theory
was
achieved
essen-
tially
by
means
of
a
thorough
study
of
the
original
papers
[IUTchI-IV],
followed
by
a
period
of
constructive
discussions
with
one
or
more
existing
experts
that
typically
lasted
roughly
from
two
to
six
months
to
sort
out
and
resolve
various
“bugs”
in
the
mathematician’s
understanding
of
the
theory
that
arose
when
the
mathematician
studied
the
original
papers
on
his/her
own
[cf.
the
discussion
of
§1.4].
On
the
other
hand,
there
is
also
a
growing
collection
of
mathematicians
who
have
a
somewhat
inaccurate
and
incomplete
—
and
indeed
often
quite
superficial
—
understanding
of
certain
aspects
of
the
theory.
This
in
and
of
itself
is
not
problematic
—
that
is
to
say,
so
long
as
the
mathematician
in
question
maintains
an
appropriate
level
of
self-awareness
of
the
inaccurate
and
incomplete
nature
of
his/her
level
of
understanding
of
the
theory
—
and
indeed
is
a
phenomenon
that
often
occurs
as
abstract
mathematical
theories
are
disseminated.
Unfortunately,
however,
a
certain
portion
of
this
collection
of
mathematicians
[i.e.,
whose
understanding
of
the
theory
is
inaccurate
and
incomplete]
have
exhibited
a
tendency
to
assert/justify
the
validity
of
their
inaccurate
and
incomplete
understand-
ing
of
the
theory
by
means
of
“reformulations”
or
“simplifications”
of
the
theory,
which
are
in
fact
substantively
different
from
and
have
no
directly
logical
relationship
to
[e.g.,
are
by
no
means
“equivalent”
to!]
the
original
theory.
Indeed,
the
version,
referred
to
in
the
present
paper
as
“RCS-IUT”
[cf.
§1.2],
that
arises
from
implementing
the
assertions
of
the
RCS
appears
to
be
the
most
famous
of
these
fabricated
versions
of
inter-universal
Teichmüller
theory
[cf.
also
the
discussion
of
Example
2.4.5
below
for
a
more
detailed
discussion
of
various
closely
related
variants
of
RCS-IUT].
On
the
other
hand,
other,
less
famous
fabricated
versions
of
inter-universal
Teichmüller
theory
have
also
come
to
my
attention
in
recent
years.
Here,
before
proceeding,
we
note
that,
in
general,
reformulations
or
simplifica-
tions
of
a
mathematical
theory
are
not
necessarily
problematic,
i.e.,
so
long
as
they
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
23
are
indeed
based
on
a
thorough
and
accurate
understanding
of
the
original
theory
and,
moreover,
can
be
shown
to
have
a
direct
logical
relationship
to
the
original
theory.
The
authoring
of
fabricated
versions
of
inter-universal
Teichmüller
theory
ap-
pears
to
be
motivated,
to
a
substantial
extent,
by
a
deep
desire
to
recast
inter-
universal
Teichmüller
theory
in
a
“simplified”
form
that
is
much
closer
to
the
sort
of
mathematics
with
which
the
author
of
the
fabricated
version
is
already
famil-
iar/feels
comfortable.
On
the
other
hand,
this
phenomenon
of
producing
fabricated
versions
also
appears
to
have
been
substantially
fueled
by
numerous
grotesquely
distorted
mass
media
reports
and
comments
on
the
English-language
internet
that
blithely
paint
inter-
universal
Teichmüller
theory
as
a
sort
of
cult
religion,
fanatical
political
movement,
mystical
philosophy,
or
vague
sketch/proposal
for
a
mathemat-
ical
theory.
Moreover,
another
unfortunate
tendency,
of
which
RCS-IUT
is
perhaps
the
most
egregious
example,
is
for
fabricated
versions
of
inter-universal
Teichmüller
theory
to
spawn
lurid
social/political
dramas
revolving
around
the
content
of
the
fabricated
version,
which
in
fact
have
essentially
nothing
to
do
with
the
content
of
inter-universal
Teichmüller
theory.
Such
lurid
dramas
then
spawn
further
grotesquely
distorted
mass
media
reports
and
comments
on
the
English-language
internet,
which
then
reinforce
and
enhance
the
social/political
status
of
the
fabricated
version
[cf.
the
discussion
of
Example
1.5.2].
Here,
it
should
be
emphasized
that
such
vicious
spirals
have
little
[or
nothing]
to
do
with
substantive
mathematical
content
and
indeed
serve
only
to
mass-produce
unnecessary
confusion
that
is
entirely
counterproductive,
from
the
point
of
the
view
of
charting
a
sound,
sustainable
course
in
the
future
development
of
the
field
of
mathematics
[cf.
the
discussion
of
[Alien],
§4.4,
(iv)].
In
fact,
of
course,
inter-universal
Teichmüller
theory
is
neither
a
religion,
nor
a
political
movement,
nor
a
mystical
philosophy,
nor
a
vague
sketch/proposal
for
a
mathematical
theory.
Rather,
it
should
be
emphasized
that
inter-universal
Teichmüller
theory
is
a
rigorously
formulated
mathematical
theory
that
has
been
verified
countless
times
by
quite
a
number
of
mathe-
maticians,
has
undergone
an
exceptionally
thorough
seven
and
a
half
year
long
refereeing
process,
and
was
subsequently
published
in
a
leading
inter-
national
journal
in
the
field
of
mathematics.
[cf.
the
discussion
of
§1.1].
In
particular,
in
order
to
avoid
the
sort
of
vicious
spirals
referred
to
above,
it
is
of
the
utmost
importance
to
concentrate,
in
discussions
of
inter-universal
Teichmüller
theory,
on
substantive
mathematical
content,
as
opposed
to
non-mathematical
—
such
as
social,
political,
or
psychological
—
aspects
or
interpretations
of
the
situation.
As
discussed
in
§1.2,
this
is
the
main
reason
for
the
use
of
the
term
“RCS”
in
the
present
paper.
24
SHINICHI
MOCHIZUKI
§1.9.
Geographical
vs.
mathematical
proximity
Historically,
mathematical
interaction
between
professional
mathematicians
re-
lied
on
physical
meetings
or
the
exchange
of
hardcopy
documents.
Increasingly,
however,
advances
in
information
technology
have
made
it
possible
for
mathemati-
cal
interaction
between
professional
mathematicians
to
be
conducted
electronically,
by
means
of
e-mail
or
online
video
communication.
Of
course,
this
does
not
imply
that
physical
meetings
or
the
exchange
of
hardcopy
documents
—
especially
in
sit-
uations
where
physical
meetings
or
the
exchange
of
hardcopy
documents
do
indeed
function
in
a
meaningful
way,
from
the
point
of
view
of
those
involved
—
should
necessarily
be
eschewed.
On
the
other
hand,
physical
meetings
between
participants
who
live
in
dis-
tant
regions
requires
travel.
Moreover,
travel,
depending
on
the
situations
of
the
participants,
can
be
a
highly
taxing
enterprise.
Indeed,
travel,
as
well
as
lodging
accommodations,
typically
requires
the
expenditure
of
a
quite
substantial
amount
of
money,
as
well
as
physical
and
mental
effort
on
the
part
of
those
involved.
This
effort
can
easily
climb
to
unmanageable
[i.e.,
from
the
point
of
view
of
certain
of
the
participants]
proportions,
especially
when
substantial
cultural
—
i.e.,
either
in
mathematical
or
in
non-mathematical
culture,
or
in
both
—
differences
are
involved.
The
current
situation
involving
the
COVID-19
pandemic
adds
yet
another
dimen-
sion
to
the
reckoning,
from
the
point
of
view
of
the
participants,
of
the
physical
and
mental
effort
that
must
be
expended
in
order
to
travel.
As
a
result,
when,
from
the
point
of
view
of
at
least
one
of
the
key
participants,
the
amount
of
effort,
time,
and/or
money
that
must
be
expended
to
travel
clearly
exceeds,
by
a
substantial
margin
—
i.e.,
“≫”
—
the
gain
[i.e.,
relative
to
various
mathematical
or
non-mathematical
criteria
of
the
key
participant
in
question]
that
appears
likely
to
be
obtained
from
the
travel
under
consideration,
it
is
highly
probable
that
the
travel
under
considera-
tion
will
end
up
simply
not
taking
place.
One
“classical”
example
of
this
phenomenon
“≫”
is
the
relative
scarcity
of
profes-
sional
mathematicians
in
Europe
or
North
America
who
travel
to
Japan
frequently
[e.g.,
at
least
once
a
year]
or
for
substantial
periods
of
time.
I
have,
at
various
times
in
my
career,
been
somewhat
surprised
by
assertions
on
the
part
of
some
mathematicians
to
the
effect
that
travel
should
somehow
be
forced
on
mathematicians,
i.e.,
to
the
effect
that
some
sort
of
coercion
may
somehow
“override”
the
fundamental
inequality
“≫”
that
exists
as
a
result
of
the
circum-
stances
in
which
a
mathematician
finds
him/herself
in.
In
my
experience,
although
this
sort
of
coercion
to
travel
may
result
in
some
sort
of
superficial
influence
in
the
very
short
term,
it
can
never
succeed
in
the
long
term.
That
is
to
say,
the
fundamental
circumstances
that
give
rise
to
the
fundamental
inequality
“≫”
can
never
be
altered
by
means
of
such
coercive
measures
to
travel
[cf.
the
discussion
of
[Rpt2014],
(8)].
In
this
context,
I
was
most
impressed
by
the
following
two
concrete
examples,
which
came
to
my
attention
recently.
In
describing
these
examples,
I
have
often
used
somewhat
indirect
expressions,
in
order
to
protect
the
privacy
of
the
people
involved.
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
25
Example
1.9.1:
The
insufficiency
of
geographical
proximity.
This
exam-
ple
concerns
the
results
obtained
in
a
paper
written
in
the
fall
of
2019
by
a
graduate
student
(St1)
from
country
(Ct1).
This
student
(St1)
showed
his
paper
to
a
promi-
nent
senior
researcher
(Pf1)
at
a
university
in
country
(Ct2)
in
a
certain
area
of
number
theory.
The
education
and
career
of
this
researcher
(Pf1)
was
conducted
entirely
at
universities
in
countries
(Ct2),
(Ct3),
and
(Ct4).
This
researcher
(Pf1)
informed
(St1)
of
his
very
positive
evaluation
of
the
originality
of
the
results
ob-
tained
in
the
paper
by
(St1).
Another
prominent
senior
researcher
(Pf2)
in
a
certain
area
of
number
theory
was
informed
by
(Pf1)
of
the
paper
by
(St1).
This
researcher
(Pf2),
who
works
at
a
university
in
country
(Ct2)
in
close
physical
proximity
to
(Pf1),
also
took
a
generally
positive
position
with
regard
to
the
paper
by
(St1).
On
the
other
hand,
several
months
subsequent
to
this
interaction
between
(Pf1)
and
(St1),
a
junior
researcher
(Pf3),
who
is
originally
from
country
(Ct5),
but
cur-
rently
works
at
a
university
in
country
(Ct2)
in
close
physical
proximity
to
(Pf1)
and
(Pf2),
informed
student
(St1)
[via
e-mail
contacts
between
(Pf3)
and
(St1)’s
advisor]
that
the
results
of
the
paper
by
(St1)
are
in
fact
“well-known”
and
essen-
tially
contained
in
papers
published
in
the
1990’s
by
(Pf4),
a
prominent
senior
researcher
in
country
(Ct6).
[To
be
more
precise,
in
fact
the
results
of
the
paper
by
(St1)
are
not
entirely
contained
in
the
papers
by
(Pf4)
in
the
sense
that
the
paper
by
(St1)
contains
certain
numerically
explicit
estimates
that
are
not
contained
in
the
papers
by
(Pf4).]
Country
(Ct6)
is
in
close
physical
proximity
to
country
(Ct3),
and
in
fact,
one
of
the
research
advisors
of
researcher
(Pf4),
when
(Pf4)
was
a
graduate
student,
was
a
prominent
researcher
(Pf5)
who
is
originally
from
country
(Ct7),
but
has
pursued
his
career
as
a
mathematician
mainly
in
countries
(Ct3)
and
(Ct6).
Here,
it
should
be
pointed
out
that
(Pf1),
(Pf2),
and
(Pf4)
are
very
close
in
age,
and
that
(Pf1)
received
his
undergraduate
education
in
country
(Ct3)
at
one
of
the
universities
that
played
in
prominent
role
in
the
career
of
(Pf5).
The
paper
by
student
(St1)
concerns
mathematics
that
has
been
studied
extensively
by
—
and
indeed
forms
one
of
the
central
themes
of
the
research
of
—
both
(Pf1)
and
(Pf4),
but
from
very
different
points
of
view,
using
very
different
techniques,
since
(Pf1)
and
(Pf4)
work
in
substantially
different
areas
of
number
theory.
On
the
other
hand,
at
no
time
during
the
initial
several
months
of
interaction
between
(Pf1),
(Pf2),
and
(St1)
was
the
work
of
(Pf4)
mentioned.
That
is
to
say,
(Pf1)
and
(Pf2)
discussed
the
results
obtained
in
the
paper
by
(St1)
in
a
way
that
can
only
be
explained
by
the
hypothesis
that
(Pf1)
and
(Pf2)
were,
at
the
time,
entirely
unaware
of
the
very
close
relationship
between
the
results
obtained
in
the
paper
by
(St1)
and
the
papers
in
the
1990’s
by
(Pf4).
—
i.e.,
despite
the
numerous
opportunities
afforded
by
close
physical
prox-
imity,
as
well
as
proximity
of
age,
for
substantial
interaction
between
(Pf1)
and
(Pf4).
The
paper
by
student
(St1)
is
currently
submitted
for
publication
to
a
cer-
tain
mathematical
journal.
Student
(St1)
recently
received
a
referee’s
report
for
his
paper,
which
apparently
[i.e.,
judging
from
the
comments
made
in
the
referee’s
report]
was
written
by
a
mathematician
working
in
an
area
of
number
theory
close
to
(Pf1).
This
referee’s
report
also
makes
no
mention
of
the
papers
in
the
1990’s
26
SHINICHI
MOCHIZUKI
by
(Pf4)
and
the
fact
that
the
results
obtained
in
the
paper
by
(St1)
appear,
with
the
exception
of
certain
numerically
explicit
estimates,
to
be
essentially
contained
in
these
papers
of
(Pf4).
Finally,
it
should
be
mentioned
that
each
official
language
of
each
of
these
countries
(Ct1),
(Ct2),
(Ct3),
(Ct4),
(Ct5),
(Ct6),
(Ct7)
belongs
to
the
European
branch
of
the
Indo-European
family
of
languages,
and
that
at
least
six
of
the
ten
pairs
of
countries
in
the
list
(Ct2),
(Ct3),
(Ct4),
(Ct6),
(Ct7)
share
a
common
official
language
[i.e.,
with
the
other
country
in
the
pair].
Example
1.9.2:
The
remarkable
potency
of
mathematical
proximity.
This
example
concerns
the
study
of
inter-universal
Teichmüller
theory
by
a
graduate
stu-
dent
(St2),
who
is
originally
from
country
(Ct8),
but
was
enrolled
in
the
doctoral
program
in
mathematics
at
a
university
in
country
(Ct9)
under
the
supervision
of
a
senior
faculty
member
(Pf6),
who
is
originally
from
country
(Ct10).
This
graduate
student
(St2)
began
his
study
of
inter-universal
Teichmüller
theory
as
a
graduate
student
and
continued
his
study
during
his
years
as
a
graduate
student
with
es-
sentially
no
mathematical
contact
with
any
researchers
who
are
significantly
involved
with
inter-universal
Teichmüller
theory,
except
for
his
advisor
(Pf6)
and
one
mid-career
researcher
(Pf7)
from
country
(Ct11).
Here,
we
remark
that
the
official
language
of
each
of
these
countries
(Ct8),
(Ct9),
(Ct10),
(Ct11)
belongs
to
the
European
branch
of
the
Indo-European
family
of
languages.
In
particular,
with
the
exception
of
a
few
very
brief
e-mail
exchanges
with
me
and
a
brief
two-
week
long
stay
at
RIMS
in
2016
to
participate
in
a
workshop
on
IUT,
this
student
(St2)
had
essentially
no
mathematical
contact,
prior
to
the
fall
of
2019,
with
any
researchers
at
Kyoto
University
who
are
involved
with
inter-universal
Teichmüller
theory.
Even
in
these
circumstances,
this
student
was
able
not
only
to
achieve
a
very
technically
sound
un-
derstanding
of
inter-universal
Teichmüller
theory
on
his
own,
by
reading
[IUTchI-IV]
and
making
use
of
various
resources,
activities,
and
contacts
within
country
(Ct9),
but
also
to
succeed,
as
a
graduate
student,
in
making
highly
nontrivial
original
research
contributions
to
a
certain
mild
generalization
of
inter-universal
Teichmüller
theory,
as
well
as
to
certain
related
aspects
of
anabelian
geometry.
My
first
[i.e.,
with
the
exception
of
a
few
very
brief
e-mail
exchanges
prior
to
this]
mathematical
contact
with
this
student
(St2)
was
in
the
fall
of
2019.
Although
this
student
(St2)
initially
had
some
technical
questions
concerning
aspects
of
inter-
universal
Teichmüller
theory
that
he
was
unable
to
understand
on
his
own,
after
a
few
relatively
brief
discussions
in
person
with
me,
he
was
able
to
find
answers
to
these
technical
questions
in
a
relatively
short
period
of
time
[roughly
a
month
or
two]
without
much
trouble.
§1.10.
Mathematical
intellectual
property
rights
The
socio-political
dynamics
generated
by
the
proliferation
of
logically
unrelated
fabricated
versions
of
inter-universal
Teichmüller
theory
—
of
which
RCS-IUT
is
perhaps
the
most
frequently
cited
[cf.
the
discussion
of
§1.2,
as
well
as
Examples
2.4.5,
2.4.7,
2.4.8
below]
—
and
further
fueled
by
·
grotesquely
distorted
mass
media
coverage
and
internet
comments
[cf.
the
discussion
of
§1.8],
as
well
as
by
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
27
·
the
conspicuous
absence
of
detailed,
explicit,
mathematically
substantive,
and
readily
accessible
written
expositions
of
the
logical
structure
underly-
ing
the
various
central
assertions
of
the
proponents
of
such
socio-political
dynamics
[cf.
the
discussion
of
§1.5],
have
had
the
effect
of
deeply
disrupting
the
normal
process
of
absorption
of
inter-universal
Teichmüller
theory
by
the
worldwide
mathematical
community.
Left
unchecked,
this
state
of
affairs
threatens
to
pave
the
way
for
a
field
—
i.e.,
the
field
of
mathematics
—
governed
by
socio-political
dynamics
[cf.
the
discussion
of
(CmSn)
in
§1.5],
rather
than
by
mathematical
content.
From
a
historical
point
of
view,
various
forms
of
institutional
and
concep-
tual
infrastructure
—
such
as
the
notions
of
·
a
modern
judiciary
system;
·
universal,
inalienable
human
rights;
·
the
rule
of
law;
·
due
process
of
law;
and
·
burden
of
proof
—
were
gradually
developed,
precisely
with
the
goal
of
averting
the
outbreak
of
the
sort
of
socio-political
dynamics
that
were
viewed
as
detrimental
to
society.
In
this
context,
it
is
interesting
to
note
the
central
role
played,
for
instance,
in
courts
of
law,
by
the
practice
of
producing
detailed,
explicit,
logically
substantive,
and
readily
accessible
written
doc-
umentation
of
the
logical
structure
underlying
the
various
central
as-
sertions
of
the
parties
involved.
This
situation
is
very
much
reminiscent
of
the
situation
in
mathematics
discussed
in
§1.5
[cf.
also
the
discussion
of
RCS-IUT
in
§1.3!],
i.e.,
where
we
observe
that
it
is
not
even
possible
to
analyze
or
debate,
in
any
sort
of
meaningfully
definitive
way,
mathematical
assertions
—
such
as,
for
instance,
the
historically
famous
assertion
of
Fermat
to
the
effect
that
he
had
a
proof
of
“Fermat’s
Last
Theorem”,
but
did
not
write
it
down
—
in
the
absence
of
such
written
documentation
of
the
logical
structure
of
the
issues
under
consideration.
From
the
point
of
view
of
the
above
discussion,
it
seems
natural,
in
the
case
of
mathematics,
to
introduce,
especially
in
the
context
of
issues
such
as
the
one
discussed
above
involving
logically
unrelated
fabricated
versions
of
inter-universal
Teichmüller
theory,
the
notion
of
mathematical
intellectual
property
rights
[i.e.,
“MIPRs”].
As
the
name
suggests,
this
notion
is,
in
some
sense,
modeled
on
the
conventional
notion
of
intellectual
property
rights
associated,
for
instance,
with
trademarks
or
brand
names
of
corporations.
In
the
case
of
this
conventional
notion,
intellectual
property
rights
may
be
understood
as
a
tool
for
protecting
the
“reli-
ability”
or
“creditworthiness”
of
trademarks
or
brand
names
of
a
corporation
from
the
sort
of
severe
injury
to
such
trademarks
or
brand
names
that
may
ensue
from
the
proliferation
of
shoddy
third-party
imitations
of
products
produced
by
the
corporation.
Here,
we
observe
that
this
“severe
injury”
often
revolves
around
the
creation
of
severe
obstacles
to
the
execution
of
activities
that
play
a
central
role
in
the
operational
normalcy
of
the
corporation.
Unlike
this
conventional
notion,
MIPRs
should
be
understood
as
being
as-
sociated
—
not
to
corporations
or
individuals
for
some
finite
period
of
time,
but
28
SHINICHI
MOCHIZUKI
rather
—
to
mathematical
notions
and
theories
and,
moreover,
are
of
unlim-
ited
duration.
The
purpose
of
MIPRs
may
be
understood
as
the
protection
of
the
“creditworthiness”
of
such
a
mathematical
notion
or
theory
from
the
severe
injury
to
the
operational
normalcy
of
mathematical
progress
related
to
no-
tion/theory
that
ensues
from
the
proliferation
of
logically
unrelated
fabricated
“fake”
versions
of
the
notion/theory.
Before
proceeding,
we
pause
to
consider
one
relatively
elementary
example
of
this
notion
of
MIPRs.
Example
1.10.1:
The
Pythagorean
Theorem.
(i)
Recall
the
Pythagorean
Theorem
concerning
the
length
of
the
hypotenuse
of
a
right
triangle
in
the
Euclidean
plane.
Thus,
if
0
<
x
≤
y
<
z
∈
R
are
the
lengths
of
the
sides
of
a
right
triangle
in
the
Euclidean
plane,
then
the
Pythagorean
Theorem
states
that
x
2
+
y
2
=
z
2
.
Various
versions
of
this
result
apparently
may
be
found
not
only
in
the
writings
of
ancient
Greece
and
Rome,
but
also
in
Babylonian,
ancient
Indian,
and
ancient
Chinese
documents.
Well-known
“elementary
proofs”
of
this
result
may
be
ob-
tained,
for
instance,
by
computing,
in
various
equivalent
ways,
the
area
of
suitable
planar
regions
covered
by
right
triangles
or
squares
that
are
closely
related
to
the
given
right
triangle.
On
the
other
hand,
such
“elementary
proofs”
typically
do
not
address
the
fundamental
issue
of
how
to
define
such
notions
as
length,
angle,
and
rotation,
i.e.,
which
are
necessary
in
order
to
understand
the
precise
content
of
the
statement
of
the
Pythagorean
Theorem.
Here,
we
observe
that
if,
for
instance,
one
tries
to
define
the
notion
of
the
length
of
a
line
segment
in
Euclidean
space
in
the
conventional
way,
then
the
Pythagorean
Theorem
reduces,
in
effect,
to
a
meaning-
less
tautology!
Moreover,
although
the
notions
of
length
and
angle
may
be
defined
once
one
has
defined
the
notion
of
a
rotation,
it
is
by
no
means
clear
how
to
give
a
natural
definition
of
the
notion
of
a
rotation.
For
instance,
one
may
attempt
to
define
the
notion
of
a
rotation
of
Euclidean
space
as
an
element
of
the
group
generated
by
well-known
matrices
involving
sines
and
cosines,
but
it
is
by
no
means
clear
that
such
a
definition
is
“natural”
or
the
“right
definition”
in
some
meaningful
sense.
Thus,
in
summary,
it
is
by
no
means
clear
that
such
“elementary
proofs”
may
be
regarded
as
genuine
rigorous
proofs
in
the
sense
of
modern
mathematics.
(ii)
From
a
modern
point
of
view,
a
natural,
precise
definition
of
the
funda-
mental
notion
of
a
rotation
of
Euclidean
space
[from
which,
as
observed
in
(i),
natural
definitions
of
the
notions
of
the
notions
of
length
and
angle
may
be
easily
derived]
may
be
given
by
thinking
in
terms
of
invariant
tensor
forms
associ-
ated
to
compact
subgroups
of
[the
topological
groups
determined
by]
various
general
linear
groups.
From
this
modern
point
of
view,
the
precise
form
of
the
Pythagorean
Theorem
[i.e.,
“x
2
+
y
2
=
z
2
”]
—
and,
in
particular,
the
significance
of
the
“2”
in
the
exponent!
—
may
be
traced
back
to
the
theory
of
Brauer
groups
and
the
closely
related
local
class
field
theory
of
the
archimedean
field
“R”
of
real
numbers,
i.e.,
in
short,
to
various
fundamental
properties
of
the
arithmetic
of
the
topological
field
“R”.
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
29
(iii)
Considering
the
situation
discussed
in
(i),
(ii),
it
is
by
no
means
clear,
in
any
sort
of
a
priori
or
naive
sense,
just
why
the
Pythagorean
Theorem
should
take
the
precise
form
“x
2
+y
2
=
z
2
”.
Indeed,
since
this
precise
form
of
the
Pythagorean
Theorem
continues
to
appear
utterly
mysterious
even
to
numerous
modern-day
high
school
students
—
i.e.,
who
grow
up
immersed
in
an
environment
replete
with
countless
cultural
links
to
modern
mathematics,
science,
and
technology!
—
it
seems
reasonable
to
assume
that
it
should
have
appeared
all
the
more
mysterious
to
the
individuals
who
populated
the
various
ancient
civilizations
mentioned
in
(i).
In
particular,
it
is
by
no
means
unnatural
to
consider
the
possibility
that
assertions
similar
to
the
following
assertions
[stated
relative
to
the
notation
introduced
in
(i)]
might
have
been
made
by
some
hypothetical
individual
at
some
time
in
human
history:
(Pyth1)
“I
don’t
understand
why
the
relation
in
the
Pythagorean
Theorem
is
of
the
form
‘x
2
+
y
2
=
z
2
’,
rather
than
‘x
2
·
y
2
=
z
2
’.”
(Pyth2)
“I
would
like
to
investigate,
in
the
context
of
the
Pythagorean
Theorem,
whether
or
not
the
relation
‘x
2
·
y
2
=
z
2
’
holds.”
(Pyth3)
“I
investigated,
in
the
context
of
the
Pythagorean
Theorem,
whether
or
not
the
relation
‘x
2
·
y
2
=
z
2
’
holds
and
discovered
that
there
exist
examples
that
show
that
this
relation
does
not
in
fact
hold
in
general.”
(Pyth4)
“The
Pythagorean
Theorem
is
false
for
the
following
reason:
The
Pythagorean
Theorem
states
that
‘x
2
·
y
2
=
z
2
’,
but
there
exist
coun-
terexamples
that
show
that
this
relation
does
not
hold
in
general.”
(iv)
From
the
point
of
view
of
the
discussion
given
above
of
MIPRs,
the
asser-
tions
(Pyth1),
(Pyth2),
(Pyth3)
do
not
constitute
a
violation
of
the
MIPRs
of
the
Pythagorean
Theorem,
but
rather
are
precisely
the
sorts
of
assertions/comments
that
occur
naturally
in
normal,
sound
research
and
educational
activities
in
math-
ematics.
By
contrast,
(VioMIPR)
(Pyth4)
may
be
regarded
as
a
classical
example
of
a
violation
of
the
MIPRs
of
the
Pythagorean
Theorem.
It
is
not
difficult
to
imagine
(DtrVio)
the
deeply
detrimental
effects
on
the
development
of
mathematics
throughout
history
that
would
have
occurred
if
violations
of
MIPRs
similar
to
(Pyth4)
regarding
the
Pythagorean
Theorem
arose
and
were
left
unchecked.
Moreover,
in
this
context,
it
is
important
to
observe
that
(IgNoJst)
assertions
of
ignorance
of
the
technical
details
that
one
must
un-
derstand
in
order
to
distinguish
the
modified
version
of
the
Pythagorean
Theorem
given
in
(Pyth4)
from
the
original
version
of
the
Pythagorean
Theorem
do
not
by
any
means
constitute
a
justification
for
participating
in
the
proliferation/citation/dissemination
of
(Pyth4).
That
is
to
say,
(BurPrf)
the
burden
of
proof
of
establishing
any
sort
of
logical
relationship
between
such
a
modified
version
and
the
original
version
lies
exclusively
in
the
hands
of
the
proponents
of
the
modified
version
30
SHINICHI
MOCHIZUKI
[cf.
the
discussion
immediately
following
the
present
Example
1.10.1].
Indeed,
this
notion
of
burden
of
proof
(BurPrf)
constitutes
a
fundamental
pillar
underlying
the
notion
of
MIPRs
and
may
be
readily
understood
by
considering
the
corre-
sponding
[perhaps
more
familiar]
situation
surrounding
the
conventional
notion
of
intellectual
property
rights
as
it
is
typically
applied
to
technological
devices
such
as
computers:
that
is
to
say,
an
assertion
of
technical
ignorance
concerning
the
details
of
the
internal
technical
structure
of
a
computer
product
manufactured
by
company
A
[or
“A-product”
for
short]
and
a
computer
product
manufactured
by
company
X
[or
“X-product”
for
short]
does
not
by
any
means
[i.e.,
legal,
ethical,
or
otherwise!]
justify
the
sale,
by
a
[say,
technically
ignorant]
computer
dealer,
of
an
X-product
advertised
as
an
authentic
A-product.
(v)
From
a
historical
point
of
view,
it
appears
to
the
author,
in
light
of
the
discussion
of
(i),
(ii),
(iii),
(iv),
to
be
in
some
sense
a
sort
of
miracle
that
the
Pythagorean
Theorem
was
“discovered”
in
and,
moreover,
survived
throughout
the
duration
of
numerous
ancient
civilizations,
i.e.,
despite
the
fact
that
dictato-
rial,
authoritarian
political
regimes
with
little
regard
for
such
modern
notions
as
a
judiciary
system
[in
the
modern
sense],
inalienable
human
rights,
the
rule
of
law,
due
process
of
law,
burden
of
proof,
and
so
on
were
by
no
means
a
rarity
in
the
ancient
world.
Of
course,
to
a
certain
extent,
this
situation
may
be
understood
as
a
consequence
of
the
fact
that
the
Pythagorean
Theorem
is
closely
related
to
the
task
of
direct
measurement
of
lengths
of
various
easily
accessed
[i.e.,
even
in
the
ancient
world!]
physical
objects.
From
this
point
of
view
of
“direct
measurement”,
the
“Pythagorean
Theorem”,
as
understood
in
various
ancient
civilizations,
should
perhaps
be
regarded
[cf.
the
discussion
of
(i),
(ii)]
not
so
much
as
a
result
in
math-
ematics
[in
the
modern
sense
of
the
term],
but
rather
as
a
principally
empirically
substantiated
result
in
physics.
Nevertheless,
even
when
viewed
from
this
point
of
view,
it
still
seems
like
something
of
a
miracle
that
this
result
survived
throughout
the
duration
of
numerous
ancient
civilizations,
unaffected
by
numerous
meaning-
less
misunderstandings
of
the
sort
discussed
in
(iii),
(iv),
especially
considering
that
similarly
meaningless
misunderstandings
of
the
Pythagorean
Theorem
continue
to
plague
modern-day
high
school
students!
Prior
to
the
introduction
above
of
the
notion
of
MIPRs,
this
notion
does
not
appear
to
have
played
an
important
role,
at
least
in
any
sort
of
explicit
sense,
in
discussions
or
analyses
of
the
development
of
mathematics.
On
the
other
hand,
Example
1.10.1
[cf.,
especially,
Example
1.10.1,
(iv)!]
shows
how,
even
at
a
purely
implicit
level,
this
notion
of
MIPRs
has
in
fact
played
a
fundamentally
important
role
in
the
development
of
mathematics
throughout
history.
Of
course,
in
the
case
of
violations
of
the
MIPRs
of
a
mathematical
notion
or
theory,
conventional
courts
or
judiciary
systems
are
simply
not
equipped
to
play
a
meaningful
role
in
dealing
with
such
violations,
since
this
would
require
an
exten-
sive
technical
knowledge
and
understanding,
on
the
part
of
the
judges
or
lawyers
involved,
of
the
mathematics
under
consideration.
Indeed,
it
is
useful
to
recall
in
this
context
that
·
traditionally,
any
detrimental
effects
arising
from
such
violations
of
the
MIPRs
of
a
mathematical
notion
or
theory
were
typically
averted
in
the
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
31
field
of
mathematics
by
means
of
the
refereeing
systems
of
various
well
established
mathematical
journals;
·
from
the
point
of
view
of
such
traditional
refereeing
systems
of
well
established
mathematical
journals,
the
burden
of
proof
regarding
the
correctness
of
any
novel
assertions
concerning
existing
mathematical
no-
tions
or
theories
—
such
as,
for
instance,
any
assertions
concerning
some
sort
of
logical
relationship
between
a
modified
version
of
a
theory
and
the
original
version
of
the
theory
—
lies
[not
with
the
author
of
the
original
version
of
the
theory
(!),
but
rather]
with
the
author
of
the
manuscript
containing
the
novel
assertions.
On
the
other
hand,
as
discussed
in
§1.8
[cf.
also
the
discussion
of
§1.3],
in
the
case
of
the
quite
egregious
MIPRs
violations
constituted
by
logically
unrelated
fabricated
versions
of
inter-universal
Teichmüller
theory,
numerous
mass
media
re-
ports
and
internet
comments
released
by
individuals
who
are
clearly
not
operating
on
the
basis
of
a
solid,
technically
accurate
understanding
of
the
mathematics
in-
volved
are
regarded,
in
certain
sectors
of
the
mathematical
community,
as
carrying
much
more
weight
than
an
exceptionally
thorough
refereeing
process
in
a
well
es-
tablished
mathematical
journal
by
experts
on
the
mathematics
under
consideration
[cf.
the
discussion
of
Example
1.5.2].
This
state
of
affairs
is
deeply
regrettable
and
should
be
regarded
as
a
cause
for
alarm.
Perhaps
in
the
long
term,
new
forms
of
institutional
or
conceptual
infrastructure
may
be
developed
for
averting
the
deeply
detrimental
effects
of
this
sort
of
situation.
At
the
time
of
writing,
however,
it
appears
that
the
only
meaningful
technical
tool
currently
available
to
humanity
for
deal-
ing
with
this
sort
of
situation
lies
in
the
production
of
detailed,
explicit,
mathematically
substantive,
and
readily
accessible
written
expositions
of
the
logical
structure
underlying
the
assertions
of
the
various
parties
involved
[cf.
the
discussion
surrounding
(OvDlk)
in
§1.5,
as
well
as
the
discussion
of
§1.12
below],
i.e.,
even
when
such
assertions
are
purported
to
be
a
“matter
of
course”
or
“com-
mon
sense”,
that
is
to
say,
a
matter
that
is
so
profoundly
self-evident
that
any
“decent,
reasonable
observer”
would
undoubtedly
find
such
written
documentations
of
logical
structure
to
be
entirely
unnecessary
[cf.
the
discussion
surrounding
(CmSn)
in
§1.5].
From
a
historical
point
of
view,
such
written
documentations
of
logical
structure
can
then
serve
as
a
valuable
transgenerational
or
transcultural
common
core
of
scholarly
activity
—
a
point
of
view
that
is
reminiscent
of
the
logical
relator
AND
“∧”,
which
forms
a
central
theme
of
the
present
paper.
§1.11.
Social
mirroring
of
mathematical
logical
structure
Discussions,
on
the
part
of
some
observers,
concerning
the
situation
surround-
ing
inter-universal
Teichmüller
theory
are
often
dominated
by
various
mutually
exclusive
and
socially
divisive/antagonistic
dichotomies
[cf.
also
the
related
discussion
of
§1.8],
i.e.,
such
as
the
following:
32
SHINICHI
MOCHIZUKI
(ExcDch)
Is
it
the
case
that
adherents
of
the
RCS
should
be
regarded
as
math-
ematically
correct/reliable/reasonable
mathematicians,
while
mathemati-
cians
associated
with
inter-universal
Teichmüller
theory
should
not,
OR
is
it
the
other
way
around?
Here,
the
“OR”
is
typically
understood
as
an
“XOR”,
i.e.,
exclusive-or.
That
is
to
say,
such
questions/dichotomies
are
typically
understood
as
issues
for
which
it
can
never
be
the
case
that
an
“AND”
relation
between
the
two
possible
alternatives
under
consideration
holds.
In
discussions
of
mutually
exclusive
dichotomies
such
as
(ExcDch),
mathematicians
associated
with
inter-universal
Teichmüller
theory,
as
well
as
the
actual
mathematical
content
of
inter-universal
Teichmüller
theory,
are
often
treated
as
completely
and
essentially
disjoint
entities,
within
the
international
mathematical
community,
from
the
mathematicians
and
mathematical
research
associated
with
the
RCS.
In
this
context,
it
is
interesting
to
note
that
this
sort
of
mutually
exclusive
dichotomy
is
very
much
reminiscent
of
the
essential
logical
structure
of
RCS-
IUT,
which,
as
discussed
in
the
latter
portion
of
Example
2.4.5
below,
may
be
understood
as
being
essentially
logically
equivalent
to
OR-IUT,
as
well
as
to
XOR-
IUT,
i.e.,
to
logically
unrelated
fabricated
versions
of
inter-universal
Teichmüller
theory
in
which
the
crucial
logical
AND
“∧”
relation
satisfied
by
the
Θ-link
of
inter-universal
Teichmüller
theory
is
replaced
by
a
logical
OR
“∨”
relation
or,
˙
relation.
alternatively,
by
a
logical
XOR
“
∨”
In
fact,
however,
(MthCnn)
although
it
is
indeed
the
case
that
the
international
mathematical
com-
munity,
as
well
as
the
mathematical
content
of
the
research
performed
by
the
international
mathematical
community,
does
not
consist
[in
the
lan-
guage
of
classical
algebraic
geometry]
of
a
single
irreducible
component,
it
is
nevertheless
undeniably
connected.
In
the
case
of
inter-universal
Teichmüller
theory,
this
abundant
inter-connectivity
may
be
explicitly
witnessed
in
the
following
aspects
of
the
theory:
(IntCnn1)
The
mathematical
content
of
various
aspects
of
inter-universal
Teichmüller
theory
is
closely
related
to
various
classical
theories
such
as
the
following:
·
the
invariance
of
heights
of
abelian
varieties
under
isogeny
[cf.
the
discussion
of
[Alien],
§2.3,
§2.4,
as
well
as
the
discussion
of
Example
3.2.1
below;
the
discussion
of
§3.5
below];
·
the
classical
proof
in
characteristic
zero
of
the
geometric
ver-
sion
of
the
Szpiro
inequality
via
the
Kodaira-Spencer
morphism,
phrased
in
terms
of
the
theory
of
crystals
[cf.
the
discussion
of
[Alien],
§3.1,
(v),
as
well
as
the
discussion
of
§3.5,
§3.10
below];
·
Bogomolov’s
proof
over
the
complex
numbers
of
the
geomet-
ric
version
of
the
Szpiro
inequality
[cf.
the
discussion
of
[Alien],
§3.10,
(vi)];
·
classical
complex
Teichmüller
theory
[cf.
the
discussion
of
Example
3.3.1
in
§3.3
below];
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
33
·
the
classical
theory
of
the
Jacobi
identity
for
the
theta
func-
tion
[cf.
the
discussion
of
Example
3.3.2
in
§3.3
below];
·
the
classical
theory
of
the
computation
of
the
Gaussian
inte-
gral
via
polar
coordinates
[cf.
[Alien],
§3.8].
We
refer
to
[Alien],
§4,
for
a
more
detailed
discussion
of
such
relationships
between
inter-universal
Teichmüller
theory
and
various
classical
mathe-
matical
theories.
(IntCnn2)
In
the
context
of
the
assertions
of
the
RCS,
it
is
important
to
recall
[cf.
the
discussion
of
(MthVl)
in
Example
2.4.5,
(viii),
below]
that
[perhaps
somewhat
surprisingly!]
in
fact
there
is
in
some
sense
no
disagreement
among
any
of
the
parties
involved
with
regard
to
the
mathematical
validity
of
the
central
mathematical
assertions
of
the
RCS
—
i.e.,
so
long
as
one
deletes
the
arbitrary
label
“inter-universal
Teichmüller
theory”
imposed
by
adherents
of
the
RCS
on
the
logi-
cally
unrelated
fabricated
versions
[i.e.,
RCS-IUT/OR-IUT/XOR-IUT]
of
inter-universal
Teichmüller
theory
that
typically
appear
in
discussions
of
adherents
of
the
RCS
[cf.
the
discussion
of
§1.2,
§1.3,
§1.8,
§1.10].
The
situation
discussed
in
(InnCnn2)
is
of
particular
interest
in
the
context
of
the
present
paper
since
the
essential
logical
structure
of
this
situation
discussed
in
(InnCnn2)
—
i.e.,
(CmMth)
of
a
common
mathematical
understanding
of
the
mathematical
va-
lidity
of
the
various
assertions
under
discussion,
so
long
as
one
keeps
track
of
the
distinct
labels
“inter-universal
Teichmüller
theory”
and
“RCS-
IUT/OR-IUT/XOR-IUT”
—
is
remarkably
similar
to
the
essential
logical
structure
of
the
situation
surrounding
the
central
theme
of
the
present
paper,
namely,
the
crucial
logical
AND
“∧”
property
of
the
Θ-link
in
inter-universal
Teichmüller
theory
—
cf.
the
discussion
of
§2.4,
§3.4
below.
At
a
more
elementary
level,
(CmMth)
may
be
understood
as
being
qualitatively
essentially
the
same
phenomenon
as
the
phenomenon
discussed
in
Example
1.10.1,
(iii),
i.e.,
the
distinction
between
(Pyth3)
[where
the
distinct
label
“Pythagorean
Theorem”
is
treated
properly]
and
(Pyth4)
[where
the
distinct
label
“Pythagorean
Theorem”
is
not
treated
properly].
§1.12.
Computer
verification,
mathematical
dialogue,
and
developmental
reconstruction
One
question
that
is
frequently
posed,
in
the
context
of
the
entirely
unnecessary
confusion
that
results
from
the
plethora
of
misinformation
and
false
narratives
concerning
inter-universal
Teichmüller
theory
in
the
English-language
mass
media
and
internet
[cf.
the
discussion
of
§1.8,
§1.10],
is
the
following:
(CmpVer)
Why
not
use
computers
to
verify
the
mathematical
validity
of
inter-
universal
Teichmüller
theory?
34
SHINICHI
MOCHIZUKI
The
implication
here
is
that
“computers”
may
be
regarded
as
an
entity
inherently
endowed
with
a
sort
of
impeccable
neutrality/impartiality
with
regard
to
the
verifi-
cation
of
mathematical
assertions.
The
fundamental
problem
with
(CmpVer)
lies
in
the
essentially
tautological
observation
that
(Algor)
no
computer
verification
algorithm
for
verifying
some
mathematical
as-
sertion
can
yield
a
verification
of
the
validity
of
the
algorithm
itself,
i.e.,
of
the
presumed
relationship
between
·
the
mechanical
output
yielded
by
the
algorithm
and
·
the
conventional
human
sense
of
“mathematical
correctness”
—
i.e.,
a
relationship
on
whose
integrity
any
sort
of
computer
verification
must
be
premised.
Of
course,
in
situations
where
the
issue
raised
in
(Algor)
is
not
an
issue
of
concern
—
i.e.,
such
as
situations
involving
routine
numerical
computations
or
manipulation
of
data
in
some
relatively
simple
combinatorial
framework
[such
as
a
finite
group,
a
finite
simplicial
complex,
or
a
finite
chain
of
Boolean
operators]
—
computer
verifi-
cation
can
indeed
function
as
a
meaningful
tool
for
the
verification
of
mathematical
assertions.
On
the
other
hand,
in
situations
—
such
as
the
situation
surrounding
such
logically
unrelated
fabricated
versions
of
inter-universal
Teichmüller
theory
as
RCS-IUT
—
in
which
the
central
issue
lies
[cf.
the
discussion
of
§1.2,
§1.3,
§1.10,
as
well
as
Example
2.4.5,
2.4.7,
2.4.8,
below]
in
the
erroneous
confusion
of
such
logically
unrelated
fabricated
versions
of
a
theory
with
the
original
version
of
the
theory,
computer
verifications
can
never
yield
meaningful
or
substantive
progress,
since
the
erroneous
confusion
of
logically
unrelated
fabricated
versions
of
a
theory
with
the
original
version
of
the
theory
completely
invalidates,
in
a
very
essential
and
inevitable
way,
the
“presumed
relationship”
discussed
in
(Algor)
on
which
any
sort
of
computer
verification
of
a
mathematical
assertion
must
be
premised.
More
elementary
examples
of
this
phenomenon
of
erroneous
confusion
of
logically
unrelated
fabricated
versions
of
a
theory
with
the
original
version
of
the
theory
may
be
seen
in
·
the
situation
surrounding
the
assertion
of
Example
1.10.1,
(iii),
(Pyth4);
·
the
situation
surrounding
the
[erroneous!]
point
of
view
described
in
Example
2.1.1
concerning
the
classical
theory
of
integration
on
the
real
line
—
i.e.,
situations
that
arise
from
essentially
non-mathematical
[e.g.,
social/political/
psychological]
circumstances
and,
as
a
result,
are
clearly
not
amenable
to
resolution
via
computer
verification.
In
the
context
of
(CmpVer),
it
is
perhaps
of
interest
to
recall
from
the
In-
troduction
[cf.
also
the
discussion
at
the
beginning
of
§3.10]
the
point
of
view
—
motivated
by
the
well-known
functional
completeness,
in
the
sense
of
propositional
calculus,
of
the
collection
of
Boolean
operators
consisting
of
logical
AND
“∧”,
logical
OR
“∨”,
and
negation
“¬”
—
that
one
can,
in
principle,
express
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
35
the
essential
logical
structure
of
any
mathematical
argument
or
theory
in
terms
of
elementary
logical
relations,
i.e.,
such
as
logical
AND
“∧”,
logical
OR
“∨”,
and
negation
“¬”.
Indeed,
it
is
precisely
this
point
of
view
that
formed
the
central
motivation
and
conceptual
starting
point
of
the
exposition
given
in
the
present
paper
concerning
the
essential
logical
structure
of
inter-universal
Teichmüller
theory,
which
may
be
represented
symbolically
as
follows:
A
∧
B
=⇒
A
∧
(B
1
∨
˙
B
2
∨
˙
.
.
.
)
A
∧
(B
1
∨
˙
B
2
∨
˙
.
.
.
∨
˙
B
1
∨
˙
B
2
∨
˙
.
.
.
)
=⇒
A
∧
(B
1
∨
˙
B
2
∨
˙
.
.
.
∨
˙
B
1
∨
˙
B
2
∨
˙
.
.
.
∨
˙
B
1
∨
˙
B
2
∨
˙
.
.
.
)
=
..
.
That
is
to
say,
in
summary:
(SymIUT)
The
symbolic
representation
[cf.
the
above
display!]
of
the
essen-
tial
logical
structure
of
inter-universal
Teichmüller
theory
exposed
in
the
present
paper
may
be
understood
as
being,
in
some
sense,
the
closest
realistic
approach
to
the
essential
spirit
of
(CmpVer).
Moreover,
this
symbolic
representation
[cf.
the
above
display!]
is
sufficiently
simple
and
transparent
that,
once
it
has
been
properly
communicated,
it
may
be
veri-
fied
readily
by
mental
computation
in
a
matter
of
minutes
without
the
use
of
a
computer!
Here,
we
note
that
the
relative
simplicity
of
this
symbolic
representation
of
(SymIUT)
is
obtained
as
a
result
of
organizing/compartmentalizing
into
“blackboxes”
vari-
ous
“blocks”
of
inter-universal
Teichmüller
theory
that
consist
of
anabelian
geometry
or
the
theory
of
étale
theta
functions,
and
whose
validity
has
never
arisen
as
a
mat-
ter
of
discussion.
Ultimately,
misunderstandings
resulting
from
logically
unrelated
fabricated
versions
can
only
be
overcome
by
studying
the
original
papers
[IUTchI-IV]
[cf.
also
[Alien],
as
well
as
the
present
paper!]
on
inter-universal
Teichmüller
theory,
or,
if
this
is
not
sufficient,
by
engaging
in
constructive
mathematical
dialogue
with
mathematicians
who
do
have
a
substantial,
accurate
understanding
of
the
theory
[cf.
the
discussion
of
§1.4,
§1.5,
§1.6].
Indeed,
perhaps
more
to
the
point,
there
appears
to
be
a
conspicuous
tendency,
in
certain
sectors
of
the
mathematical
community,
to
(RfsDlg)
utilize
the
proliferation/citation
of
logically
unrelated
fabricated
versions
of
the
theory
as
a
sort
of
“lame
excuse”/subterfuge
to
justify
a
stance
of
refusal
to
engage
in
such
constructive
mathematical
dialogue
concerning
the
theory
—
cf.
the
discussion
of
§1.3,
§1.10,
especially
the
discussion
of
(VioMIPR),
(DtrVio),
(IgNoJst),
(BurPrf)
in
Example
1.10.1,
(iv);
the
discussion
of
Examples
3.10.1,
3.10.2
below.
Moreover,
(DngPrc)
the
situation
described
above
in
(RfsDlg),
if
left
unchecked,
constitutes,
in
the
long-term,
a
dangerous
precedent
from
the
point
of
view
of
36
SHINICHI
MOCHIZUKI
maintaining
a
state
of
operational
normalcy
in
the
field
of
mathematics
in
a
fashion
consistent
with
such
fundamental
democratic
principles
as
the
rule
of
law,
due
process
of
law,
and
burden
of
proof
—
cf.
the
discussion
of
§1.10,
especially
the
discussion
of
(VioMIPR),
(DtrVio),
(IgNoJst),
(BurPrf)
in
Example
1.10.1,
(iv).
Nevertheless,
in
this
context,
it
is
also
of
fundamental
importance
to
recall
(OvrMs)
the
existence
of
numerous
mathematicians
[of
many
diverse
nationali-
ties!]
who
were
indeed
successful
in
overcoming
various
meaningless
misunderstandings
concerning
inter-universal
Teichmüller
theory
pre-
cisely
by
persistently
engaging
in
constructive
mathematical
dialogue
concerning
the
theory.
Such
dialogues
typically
involve
the
painstaking
and
time-consuming
process
of
(ExplLS)
sifting
through
and
analyzing
the
assertions
of
the
mathematician
in
question
concerning
inter-universal
Teichmüller
theory
in
order
to
make
precise
and
explicit
the
exact
content
of
the
logical
structure
under-
lying
such
assertions
[cf.
the
discussion
of
§1.5,
especially
Examples
1.5.1,
1.5.2].
On
the
other
hand,
it
is
important
to
emphasize
that
such
efforts,
when
seen
through
to
their
conclusion,
have
always
led
to
a
situation
in
which
the
mathematician
in
question
realizes
his/her
misunderstandings
of
the
theory
and
ultimately
concedes
that,
at
least
so
far
as
he/she
can
see,
(MthVlTh)
there
is
indeed
no
mathematical
reason
to
deny
the
mathematical
validity
of
the
theory.
That
is
to
say,
this
chain
of
events
is
precisely
the
chain
of
events
that
should
be
expected
from
any
constructive
mathematical
dialogue
carried
out
in
a
suitable,
sincere,
and
rational
fashion
concerning
a
rigorously
formulated
mathematical
the-
ory.
Thus,
in
summary,
at
least
in
a
direct,
literal
sense,
ultimately,
the
only
way
to
overcome
meaningless
misunderstandings
of
the
theory
that
arise
from
logically
unrelated
fabricated
versions
of
the
theory
is
to
(CfrMth)
directly
confront
the
mathematical
content
of
the
original
theory,
either
·
by
studying
the
original
papers
[IUTchI-IV]
[cf.
also
[Alien],
as
well
as
the
present
paper!]
on
inter-universal
Teichmüller
theory,
or,
if
this
is
not
sufficient,
·
by
engaging
in
constructive
mathematical
dialogue
with
mathe-
maticians
who
do
have
a
substantial,
accurate
understanding
of
the
theory.
[cf.
the
discussion
of
§1.3,
§1.10,
as
well
as
Examples
3.10.1,
3.10.2
below].
Conversely,
(BlkAcc)
a
stance
of
systematic
and
institutionally
endorsed
refusal
to
confront
the
mathematical
content
of
the
original
theory
has
the
effect
of
orches-
trating
the
creation
of
a
sort
of
artificial
blackhole
relative
to
the
issue
of
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
37
mathematical
accountability
[cf.
the
discussion
of
[EMSCOP]
in
§1.3]
and
leads
to
the
sort
of
absurd
pathologies
and
obstructions
to
the
operational
normalcy
of
the
field
of
mathematics
discussed
in
detail
in
§1.10
[cf.,
especially,
Example
1.10.1,
(iii),
(Pyth4)].
That
is
to
say,
unlike
the
IUT
community,
which
bears
active
mathematical
responsibility
for
the
mathematical
content
of
inter-universal
Teichmüller
theory
in
the
long-term
by
maintaining
an
extensive,
sustained
infrastructural
appa-
ratus
of
mathematical
activities
surrounding
inter-universal
Teichmüller
theory
—
such
as
·
workshops,
of
one
to
two
weeks
in
length,
concerning
inter-universal
Teichmüller
theory
[e.g.,
at
RIMS
in
March
2015,
in
Oxford
in
Decem-
ber
2015,
at
RIMS
in
July
2016,
and
at
RIMS
in
September
2021]
and
numerous
lecture
series
[e.g.,
in
Kumamoto
in
May
2014,
at
RIMS
in
De-
cember
2015,
in
Yokohama
in
November
2018,
and
at
RIMS
in
December
2021],
which
have
led
to
a
quite
substantial
stock
of
widely
and
readily
ac-
cessible
PDF
files
of
slides
and
videos
of
lectures
exposing
inter-universal
Teichmüller
theory;
·
extensive
one-to-one
mathematical
discussions
between
mathematicians
all
over
the
world
concerning
inter-universal
Teichmüller
theory
via
e-mail
and
online
video
meetings;
·
joint
research
projects
concerning
the
further
development
of
inter-
universal
Teichmüller
theory
[cf.,
e.g.,
[ExpEst]]
—
the
author
remains
entirely
unable,
despite
years
of
intensive
effort
[cf.
the
discussion
of
§1.3,
§1.5,
§1.6],
to
locate
even
a
single
mathematician
who
is
will-
ing
to
bear
active
responsibility
for
the
mathematical
content
of
RCS-IUT
by
engaging
in
similar
mathematical
activities/mathematical
dialogue.
The
fun-
damental
qualitative
difference
constituted
by
this
egregrious
absence
of
an
infrastructural
apparatus
supporting
the
assertions
of
the
RCS
—
that
is
to
say,
put
another
way,
this
sort
of
“hit-and-run”/“dead-end”
approach
to
making
vaguely
formulated
mathematical
assertions
that
are
not
supported
by
detailed
documentation/exposition
apparatuses,
i.e.,
in
violation
of
the
[EMSCOP]
[cf.
the
discussion
of
the
[EMSCOP]
in
§1.3]
—
is
precisely
what
is
meant
by
the
notion
of
a
“blackhole”
of
mathematical
accountability
discussed
in
(BlkAcc).
On
the
other
hand,
from
a
more
long-term,
historical
point
of
view,
it
is
perhaps
of
interest
to
observe
that
there
is
another
approach
to
witnessing
the
validity
of
a
mathematical
theory,
namely,
(DvpRcn)
the
approach
of
developmental
reconstruction,
i.e.,
of
“reconstructing
the
validity”
of
a
mathematical
theory
by
witnessing
the
subsequent
developments
that
ensue
from
the
theory.
This
point
of
view
is
particularly
of
interest
in
the
context
of
inter-universal
Te-
ichmüller
theory,
given
the
central
role
played
in
inter-universal
Teichmüller
theory
by
anabelian
geometry,
i.e.,
which
revolves
around
the
development
of
recon-
struction
algorithms
that
allow
one
to
reconstruct
conventional
algebraic
struc-
tures
[i.e.,
of
the
sort
that
typically
appear
in
algebraic/arithmetic
geometry]
from
more
primitive
combinatorial
structures
such
as
topological
groups.
38
SHINICHI
MOCHIZUKI
This
approach
of
developmental
reconstruction
may
be
applied,
for
instance,
to
the
task
of
evaluating
the
level
of
mathematical
or
scientific
development
of
ancient
civilizations,
i.e.,
not
via
the
direct
study
of
detailed
theoretical
expositions
[which
are
typically
not
readily
available
—
cf.
the
discussion
of
§1.5!]
of
the
mathematics
or
science
understood
by
such
an
ancient
civilization,
but
rather
by
observing
what
may
be
understood
as
the
“fruits”
of
this
mathematics
or
science,
e.g.,
in
the
form
of
architectural
achievements
such
as
the
famous
·
pyramids
of
Egypt
or
·
Nazca
lines
and
mysterious
ruins
of
Puma
Punku
in
South
Amer-
ica.
Another
important
[though
non-architectural!]
example
of
this
sort
of
phenomenon
may
be
seen
in
·
the
list
of
Pythagorean
triples
in
the
famous
Babylonian
tablet
Plimpton
322
—
i.e.,
which
is
particularly
notable
in
that
it
strongly
suggests
[that
is
to
say,
despite
the
fact
that
it
is
not
accompanied
by
any
sort
of
explicit
theoretical
exposition!]
an
understanding
of
algebraic
manipulation
on
a
par
with
the
essential
content
of
Example
1.12.1
below.
Example
1.12.1:
Explicit
parametrization
of
Pythagorean
triples.
The
set
of
integral
solutions
—
i.e.,
solutions
in
the
ring
of
integers
Z,
also
known
as
Pythagorean
triples
—
of
the
equation
x
2
+y
2
=
z
2
may
be
parametrized
by
applying
the
substitutions
xz
→
u,
yz
→
v
and
considering
the
set
of
rational
solutions
—
i.e.,
solutions
in
the
field
of
rational
numbers
Q
—
of
the
equation
u
2
+
v
2
=
1.
The
solutions
of
this
last
equation
u
2
+
v
2
=
1
in
Q
—
or,
indeed,
in
any
field
of
characteristic
=
2
—
may
be
completely
parametrized
by
the
substitutions
u
→
t
t
2
−1
+1
,
2
v
→
t
2
2t
+1
—
where
we
observe
that
2
t
2
−1
t
2
+1
+
2
·
2
2t
t
2
+1
=
1,
−1
=
2
·
t
+
1
t
t
=
=
u+1
v
−1
u
2
+v
2
+1+2u
v(u+1)
−1
=
=
2
·
u+1
v
v
+
u+1
=
2
·
v
1−u
=
2
·
2(u+1)
v(u+1)
t
2
−1
t
2
+1
−1
2t
,
+
1
·
t
2
+1
−1
(u+1)
2
+v
2
v(u+1)
−1
=
v
That
is
to
say,
this
parametrization
by
t
[for
t
such
that
t
2
+
1
=
0]
gives
a
complete
list
of
all
solutions
of
the
equation
u
2
+
v
2
=
1
[for
u
such
that
u
−
1
=
0]
in
any
field
of
characteristic
=
2
[or,
indeed,
by
interpreting
“
=
0”
as
a
condition
of
invertibility,
in
any
ring
in
which
2
is
invertible].
On
the
other
hand,
in
this
context,
it
is
of
interest
to
note
that,
at
least
as
of
the
time
of
writing
of
the
present
paper,
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
39
(AncBas)
no
ancient
civilization
has
produced
evidence
of
knowledge
of
the
equation
∞
1
n=1
n
2
=
π
2
6
—
i.e.,
of
the
solution
of
the
so-called
Basel
problem.
That
is
to
say,
here
we
note
that
this
observation
(AncBas)
is
valid
despite
the
fact
that
each
of
the
essential
components
of
this
equation
—
i.e.,
·
the
positive
integers
of
“sufficiently
large
value”,
·
the
elementary
operations
of
addition/multiplication/division,
·
the
notion
of
the
length
of
the
circumference
of
a
circle
of
radius
1,
and
·
the
idea
of
a
sum
of
[real]
numbers
coming
arbitrarily
close,
up
to
a
very
small
margin
of
error,
to
another
[real]
number
—
may
be
readily
expressed
in
terms
understandable
to
many
advanced
ancient
civilizations.
Of
course,
the
discovery
of
an
ancient
civilization
that
produced
evidence
of
some
sort
of
knowledge
of
the
equation
in
the
display
of
(AncBas)
would
be
quite
startling
since
it
would
strongly
suggest
[that
is
to
say,
even
if
it
is
not
accompanied
by
any
sort
of
explicit
theoretical
exposition!]
an
understanding
of
numerous
ideas
and
theorems
not
only
from
elementary
differential
and
integral
calculus,
but
also
possibly
from
Fourier
analysis
on
the
circle
and
complex
analysis
on
the
complex
plane.
Finally,
we
return
to
our
discussion
of
inter-universal
Teichmüller
theory.
In
the
case
of
inter-universal
Teichmüller
theory,
the
phenomenon
of
developmental
reconstruction
may
already
be
seen,
albeit
in
a
relatively
weak
sense,
in
the
nu-
merical
results
of
[ExpEst].
Stronger
examples
of
this
phenomenon
may
be
seen,
however,
in
various
enhanced
versions
of
inter-universal
Teichmüller
theory
that
are
currently
under
development,
which
are
expected
to
give
rise
to
various
new
types
of
applications
of
inter-universal
Teichmüller
theory.
Such
enhanced
versions
suggest
strongly
that
the
original
version
of
inter-universal
Teichmüller
theory
given
in
[IUTchI-IV]
[cf.
also
[Alien],
as
well
as
the
present
paper!]
should
perhaps
be
regarded
as
being
only
the
first
example
of
a
much
larger
collection
of
examples
of
“anabelian
adèlic
analysis”,
i.e.,
in
the
spirit
of
the
point
of
view
that
the
various
types
of
prime-strips
that
occur
in
inter-universal
Teichmüller
theory
may
be
thought
of
[cf.
the
discussion
at
the
end
of
[Alien],
§3.3,
(iv)]
as
a
sort
of
anabelian/monoid-theoretic
version
of
the
classical
notion
of
adèles/idèles
that
ap-
pears
throughout
conventional
arithmetic
geometry
and
number
theory.
Here,
we
observe
that
this
term
“anabelian
adèlic
analysis”
is
of
interest
from
a
historical
point
of
view
in
that
it
encapsulates,
in
a
perhaps
surprisingly
efficient
fashion,
a
quite
substantial
portion
of
the
historical
development
of
arithmetic
geometry
that
underlies
significant
portions
of
inter-universal
Teichmüller
theory:
indeed,
(AAA0)
the
non-holomorphic
(!)
“analysis”
on
the
real
line
surrounding
Eu-
ler’s
formula
[cf.
the
discussion
of
§1.5],
Euler’s
solution
of
the
Basel
problem
[cf.
the
discussion
above
of
(AncBas)],
the
gamma
function,
and
the
Gaussian
integral
[also
known
as
the
Euler-Poisson
integral
—
cf.
also
40
SHINICHI
MOCHIZUKI
the
discussion
of
[Alien],
§3.8]
—
i.e.,
analysis
of
the
sort
practiced
during
the
18-th
century
by
such
mathematicians
as
Euler
and
the
Bernoullis
—
may
be
understood
as
a
sort
of
essential
preparatory
phase
that
paved
the
way
for
the
holomorphic
analysis/function
theory
of
the
19-th
century
discussed
in
(AAA1),
below,
surrounding
the
theta
and
zeta
functions
[as
well
as
the
complex
logarithm
—
cf.
the
discussion
of
§1.5];
(AAA1)
at
a
more
genuine/non-preparatory
level,
“analysis”
may
be
regarded
as
referring
to
the
classical
complex
function
theory
—
pioneered
by
such
19-th
century
mathematicians
as
Jacobi,
Riemann,
and
Mellin
—
on
the
complex
plane/upper
half-plane
surrounding
the
well-known
functional
equations
of
the
theta
function
and
the
Riemann
zeta
function,
which
are
related
via
the
Mellin
transform
[cf.
the
discussion
of
§1.5];
(AAA2)
“adèlic”
may
be
understood
as
referring
to
the
adèlization
—
devel-
oped
by
such
20-th
century
mathematicians
as
Chevalley,
Weil,
Iwa-
sawa,
Artin,
Tate,
and
Langlands
—
of
the
classical
complex
function
theory
of
(AAA1),
a
development
which
led,
in
particular,
to
the
well-
known
adèlic
proof
of
the
functional
equation
of
the
Dedekind
zeta
function
[originally
due
to
Hecke]
and
subsequently
to
the
representation-theoretic
approach
of
the
Langlands
program;
(AAA3)
“anabelian”
may
be
interpreted
as
referring
to
the
“anabelianiza-
tion”
of
the
classical
functional
equation
of
the
theta
function
on
the
up-
per
half-plane
in
the
fashion
of
inter-universal
Teichmüller
theory,
i.e.,
by
relating
Galois
groups/arithmetic
fundamental
groups
to
ring/field
theory
—
not
via
representation
theory,
as
in
(AAA2)
(!),
but
rather
—
by
ap-
plying
cyclotomic
rigidity
isomorphisms
and
Kummer
theory
to
relate
the
étale-like
objects
constructed
by
means
of
anabelian
algorithms
to
their
Frobenius-like
counterparts
arising
from
ring/field
theory.
Thus,
(AAA2)
may
be
understood
as
an
approach
to
the
“arithmetization”
of
the
classical
function
theory
of
(AAA1)
by
means
of
adèlization/representation
theory,
i.e.,
by
thinking,
in
short,
of
the
adèles
as
a
new
domain
[i.e.,
more
pre-
cisely,
locally
compact
topological
space
of
uncountable
cardinality]
in
which
to
conduct
analysis/function
theory.
By
contrast,
(AAA3)
may
be
interpreted
as
an
approach
to
the
“arithmetization”
of
the
classical
function
theory
of
(AAA1)
by
thinking
of
the
abstract
topological
groups
that
arise
as
absolute
Galois
groups/arithmetic
fundamental
groups
as
the
natural
domain
[i.e.,
more
pre-
cisely,
locally
compact
topological
space
of
uncountable
cardinality]
for
conducting
analysis/function
theory.
Thus,
in
summary,
at
the
level
of
natural
domains
for
conducting
the
analysis/function
theory
surrounding
the
theta/zeta
functions,
one
can
discern
a
fascinating
historical
progression
(AAA0)
(AAA1)
(AAA2)
(AAA3)
corresponding
to
R
C
adèles
arithmetic
fundamental
groups.
In
closing,
we
note
that,
in
some
sense,
this
interpretation
of
(AAA3)
is
consistent
with
the
spirit
of
Grothendieck’s
anabelian
philosophy
as
an
approach
to
diophantine
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
41
geometry
via
anabelian
geometry,
although,
as
discussed
in
[IUTchI],
§I5,
whereas
Grothendieck
apparently
envisaged
this
approach
as
centering
around
the
Section
Conjecture,
the
anabelian
geometry
that
actually
appears
in
inter-universal
Te-
ichmüller
theory
consists
mainly
of
absolute
anabelian
geometry
over
number
fields
and
p-adic
local
fields.
Section
2:
Elementary
mathematical
aspects
of
“redundant
copies”
The
essence
of
the
central
mathematical
assertions
of
the
RCS
revolves,
per-
haps
somewhat
remarkably,
around
quite
elementary
considerations
that
lie
well
within
the
framework
of
undergraduate-level
mathematics.
Before
examining,
in
§3,
the
assertions
of
the
RCS
in
the
technical
terminology
of
inter-universal
Teichmüller
theory,
we
pause
to
give
a
detailed
exposition
of
these
elementary
considerations.
§2.1.
The
history
of
limits
and
integration
The
classical
notion
of
integration
[e.g.,
for
continuous
real-valued
functions
on
the
real
line],
as
well
as
the
more
fundamental,
but
closely
related
notion
of
a
limit,
have
a
long
history,
dating
back
[at
least]
to
the
17-th
century.
Initially,
these
notions
did
not
have
rigorous
definitions,
i.e.,
were
not
“well-defined”,
in
the
sense
understood
by
mathematicians
today.
The
lack
of
such
rigorous
definitions
frequently
led,
up
until
around
the
end
of
the
19-th
century,
to
“contradictions”
or
“paradoxes”
in
mathematical
work
—
such
as
Grandi’s
series
∞
(−1)
n
n=0
—
concerning
integrals
or
limits.
Ultimately,
of
course,
the
theory
of
limits
and
integrals
evolved,
especially
during
the
period
starting
from
around
the
mid-19-th
century
and
lasting
until
around
the
early
20-th
century,
to
the
extent
that
such
“contradictions/paradoxes”
could
be
resolved
in
a
definitive
way.
This
process
of
evolution
involved,
for
instance,
in
the
case
of
integration,
first
the
introduction
of
the
Riemann
integral
and
later
the
introduction
of
the
Lebesgue
integral,
which
made
it
possible
to
integrate
functions
—
such
as,
for
instance,
the
indicator
function
on
the
real
line
of
the
subset
of
rational
numbers
—
whose
Riemann
integral
is
not
well-defined.
Here,
it
should
be
noted
that
at
various
key
points
during
this
evolution
of
the
notions
of
limits
and
integration,
the
central
“contradictions/paradoxes”
that,
at
times,
led
to
substantial
criticism
and
confusion
arose
from
a
solid,
technically
accurate
understanding
of
the
content
and
logical
structure
of
the
assertions
—
such
as,
for
instance,
various
possible
approaches
to
computing
the
value
of
Grandi’s
series
—
at
the
center
of
these
“contradictions/paradoxes”.
It
is
precisely
for
this
reason
that
such
criticism
and
confusion
ultimately
lead
to
substantive
refinements
in
the
theory
that
were
sufficient
to
resolve
the
original
“contradictions/paradoxes”
in
a
definitive
way.
Such
constructive
episodes
in
the
history
of
mathematics
—
which
may
be
studied
by
scholars
today
precisely
because
of
the
existence
of
detailed,
42
SHINICHI
MOCHIZUKI
explicit,
mathematically
substantive,
and
readily
accessible
written
records!
[cf.
the
discussion
of
§1.5]
—
stand
in
stark
contrast
to
[cf.
the
discussion
of
(UndIg)
in
§1.3]
criticism
of
a
mathematical
theory
that
is
based
on
a
fundamental
ignorance
of
the
content
and
logical
structure
of
the
the-
ory,
such
as
the
following
“false
contradiction”
in
the
theory
of
integration,
which
may
be
observed
in
some
students
who
are
still
in
an
initial
stage
with
regard
to
their
study
of
the
notion
of
integration.
Example
2.1.1:
False
contradiction
in
the
theory
of
integration.
Consider
the
following
computation
of
the
definite
integral
of
a
real-valued
function
on
the
real
line
1
x
n
dx
=
0
1
n+1
for
n
a
positive
integer.
Suppose
that
one
takes
the
[drastically
oversimplified
and
manifestly
absurd,
from
the
point
of
view
of
any
observer
who
has
an
accurate
understanding
of
the
theory
of
integration!]
point
of
view
that
the
most
general
possible
interpretation
of
the
equation
of
the
above
display
is
one
in
which
the
following
three
symbols
1
“
”,
“x”,
“dx”
0
are
allowed
to
be
arbitrary
positive
real
numbers
a,
b,
c.
Here,
we
note
that
in
the
case
of
“dx”,
such
a
substitution
“dx
→
c”
could
be
“justified”
by
quoting
conven-
tional
“-δ”
treatments
of
the
theory
of
limits
and
integrals,
in
which
infinitesimals
—
i.e.,
such
as
“dx”
or
“”
—
are
allowed
to
be
arbitrary
positive
real
numbers,
which
are
regarded
as
being
“arbitrarily
small”,
and
observing
that
any
positive
real
number
c
is
indeed
much
smaller
than
“most
other
positive
real
numbers”
[such
as
1000
·
c,
etc.].
On
the
other
hand,
by
substituting
the
values
n
=
1,
2,
3,
one
obtains
relations
abc
=
1,
ab
2
c
=
12
,
ab
3
c
=
13
.
The
first
two
of
these
relations
imply
that
b
=
12
[so
b
2
=
14
],
while
the
first
and
third
relations
imply
that
b
2
=
13
=
14
—
a
“contradiction”!
§2.2.
Derivatives
and
integrals
In
the
context
of
the
historical
discussion
of
integration
in
§2.1,
it
is
interesting
to
recall
the
fundamental
theorem
of
calculus,
i.e.,
the
result
to
the
effect
that,
roughly
speaking,
the
operations
of
integration
and
differentiation
of
functions
[i.e.,
real-valued
functions
on
the
real
line
satisfying
suitable
conditions]
are
inverse
to
one
another.
Thus,
from
a
certain
point
of
view,
the
“essential
information”
contained
in
a
function
may
be
understood
as
being
“essentially
equivalent”
to
the
“essential
information”
contained
in
the
derivative
of
the
function
—
that
is
to
say,
since
one
may
always
go
back
and
forth
at
will
between
a
function
and
its
derivative
by
integrating
and
then
differentiating.
This
point
of
view
might
then
tempt
some
observers
to
conclude
that
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
43
any
mathematical
proof
that
relies,
in
an
essential
way,
on
consideration
of
the
derivative
of
a
function
must
be
fundamentally
flawed
since
any
information
that
might
possibly
be
extracted
from
the
derivative
of
the
function
should
already
be
available
[cf.
the
“essential
equivalence”
dis-
cussed
above]
from
the
function
prior
to
passing
to
the
derivative,
i.e.,
in
“contradiction”
to
the
essential
dependence
of
the
proof
on
passing
to
the
derivative.
Alternatively,
this
point
of
view
may
be
summarized
in
the
following
way:
the
“essential
equivalence”
discussed
above
implies
that
any
usage,
in
a
mathematical
proof,
of
the
derivative
of
a
function
is
necessarily
inherently
redundant
in
nature.
In
fact,
of
course,
such
“pseudo-mathematical
reasoning”
is
itself
fundamen-
tally
flawed.
Two
examples
of
well-known
proofs
in
arithmetic
geometry
that
depend,
in
a
essential
way,
on
passing
to
the
derivative
will
be
discussed
in
the
final
portion
of
§3.2
below.
These
examples
are
in
fact
closely
related
to
the
mathematics
that
inspired
inter-universal
Teichmüller
theory
[cf.
the
discussion
in
the
final
por-
tion
of
§3.2
below].
One
central
aspect
of
the
situations
discussed
in
§3.2
below
is
the
exploitation
of
properties
of
[various
more
abstract
analogues
of]
the
derivative
of
a
function
that
differ,
in
a
very
substantive,
qualitative
way,
from
the
prop-
erties
of
the
original
function.
One
important
example
of
this
sort
of
situation
is
the
validity/invalidity
of
various
symmetry
properties.
This
phenomenon
may
be
observed
in
the
following
elementary
example.
Example
2.2.1:
Symmetry
properties
of
derivatives.
The
real-valued
func-
tion
f
(x)
=
x
on
the
real
line
is
not
invariant
[i.e.,
not
symmetric]
with
respect
to
translations
by
an
arbitrary
constant
c
∈
R.
That
is
to
say,
in
general,
it
is
not
the
case
that
f
(x
+
c)
=
f
(x).
On
the
other
hand,
the
derivative
f
(x)
=
1
of
this
function
is
manifestly
invariant/symmetric
with
respect
to
such
trans-
lations.
§2.3.
Line
segments
vs.
loops
By
comparison
to
the
examples
given
in
§2.1,
§2.2,
the
following
elementary
geometric
examples
are
much
more
closely
technically
related
to
the
assertions
of
the
RCS
concerning
inter-universal
Teichmüller
theory.
Example
2.3.1:
Endpoints
of
an
oriented
line
segment.
(i)
Write
def
I
=
[0,
1]
⊆
R
44
SHINICHI
MOCHIZUKI
for
the
closed
unit
interval
[i.e.,
the
set
of
nonnegative
real
numbers
≤
1]
in
the
real
line
R.
Thus,
I
is
equipped
with
a
natural
topology
[i.e.,
induced
by
the
topology
of
R],
hence
can
be
regarded
as
a
topological
space,
indeed
more
specifically,
as
a
one-dimensional
topological
manifold
with
boundary
that
is
equipped
with
a
natural
orientation
[i.e.,
induced
by
the
usual
orientation
of
R].
Write
def
α
=
{0},
def
β
=
{1}
for
the
topological
spaces
[consisting
of
a
single
point!]
determined
by
the
two
endpoints
of
I.
Thus,
α
and
β
are
isomorphic
as
topological
spaces.
In
certain
situations
that
occur
in
category
theory,
it
is
often
customary
to
replace
a
given
category
by
a
full
subcategory
called
a
skeleton,
which
is
equivalent
to
the
given
category,
but
also
satisfies
the
property
that
any
two
isomorphic
objects
in
the
skeleton
are
equal.
This
point
of
view
of
working
with
skeletal
categories
[i.e.,
categories
which
are
their
own
skeletons]
is
motivated
by
the
idea
that
nonequal
isomorphic
objects
are
“redundant”.
Of
course,
there
are
indeed
various
situations
in
which
nonequal
isomorphic
objects
are
redundant
in
the
sense
that
working
with
skeletal
categories,
as
opposed
to
arbitrary
categories,
does
not
result
in
any
substantive
difference
in
the
mathematics
under
consideration.
(ii)
On
the
other
hand,
if,
in
the
present
discussion
of
I,
α,
β
—
which
one
may
visualize
as
follows
I
•——>——•
α
β
—
one
identifies
α
and
β,
then
one
obtains
a
new
topological
space,
that
is
to
say,
more
specifically,
an
oriented
one-dimensional
topological
manifold
[whose
orienta-
tion
is
induced
by
the
orientation
of
I]
def
L
=
I/
α
∼
β
that
is
homeomorphic
to
the
unit
circle,
i.e.,
may
be
visualized
as
a
loop.
Write
γ
L
⊆
L
for
the
image
of
α
⊆
I,
or,
equivalently,
β
⊆
I,
in
L.
As
is
well-known
from
elementary
topology,
the
topological
space
L
is
structurally/qualitatively
very
different
from
the
topological
space
I.
For
instance,
whereas
I
has
a
trivial
fun-
damental
group,
L
has
a
nontrivial
fundamental
group
[isomorphic
to
the
additive
group
of
integers
Z].
In
particular,
it
is
by
no
means
the
case
that
the
fact
that
α
and
β
are
isomorphic
as
topological
spaces
implies
a
sort
of
“redundancy”
to
the
effect
that
any
mathematical
argument
involving
I
[cf.
the
above
observation
concerning
fundamental
groups!]
is
entirely
equivalent
to
a
corresponding
math-
ematical
argument
in
which
α
and
β
are
identified,
i.e.,
in
which
“I”
is
replaced
by
“L”.
(iii)
In
this
context,
we
observe
that
the
[one-dimensional
oriented
topological
manifold
with
boundary]
I
does
not
admit
any
symmetries
that
switch
α
and
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
45
β.
Moreover,
even
if
one
passes
to
the
quotient
I
L,
the
[one-dimensional
ori-
ented
topological
manifold]
L
does
not
admit
any
symmetries
that
reverse
the
orientation
of
L.
Example
2.3.2:
Gluing
of
adjacent
oriented
line
segments.
(i)
A
similar
elementary
geometric
situation
to
the
situation
discussed
in
Ex-
ample
2.3.1,
but
which
is
technically
a
bit
more
similar
to
the
situation
that
arises
in
inter-universal
Teichmüller
theory
may
be
given
as
follows.
We
begin
with
two
distinct
copies
†
I,
‡
I
of
I.
Thus,
†
I
has
endpoints
†
α,
†
β
[i.e.,
corresponding
re-
spectively
to
the
endpoints
α,
β
of
I];
similarly,
‡
I
has
endpoints
‡
α,
‡
β.
We
then
proceed
to
form
a
new
topological
space
J
by
gluing
†
I
to
‡
I
via
the
unique
isomor-
∼
phism
of
topological
spaces
†
β
→
‡
α.
Thus,
†
β
and
‡
α
are
identified
in
J.
Let
us
write
γ
J
for
the
one-pointed
topological
space
obtained
by
identifying
†
β
and
‡
α.
Thus,
J
may
be
visualized
as
follows:
†
‡
I
I
•
——>——•——>——
•
†
‡
α
γ
J
β
(ii)
Observe
that
the
gluing
operation
that
gave
rise
to
J
is
such
that
we
may
regard
†
I
and
‡
I
as
subspaces
†
I
⊆
J,
‡
I
⊆
J
of
J.
Since
each
of
these
subspaces
†
I,
‡
I
of
J
is
naturally
isomorphic
to
I,
one
may
take
the
point
of
view,
as
in
the
discussion
of
Example
2.3.1,
that
these
two
subspaces
are
“redundant”
and
hence
should
be
identified
with
one
another
[say,
via
the
natural
isomorphisms
of
†
I,
†
I
with
I]
to
form
a
new
topological
space
M
=
J/
†
I
∼
‡
I
def
—
where
we
observe
that
the
natural
isomorphisms
of
†
I,
†
I
with
I
determine
a
∼
natural
isomorphism
of
topological
spaces
M
→
L
=
I/
α
∼
β,
with
the
loop
L
considered
in
Example
2.3.1.
Write
γ
M
⊆
M
for
the
image
of
γ
J
⊆
J
in
M.
Thus,
∼
the
natural
isomorphism
M
→
L
maps
γ
M
isomorphically
onto
γ
L
.
On
the
other
hand,
just
as
in
the
situation
discussed
in
Example
2.3.1,
it
is
by
no
means
the
case
that
the
fact
that
†
I
and
‡
I
are
[in
fact,
natu-
rally]
isomorphic
as
topological
spaces
implies
a
sort
of
“redundancy”
to
the
effect
that
any
mathematical
argument
involving
J
is
entirely
equiv-
alent
to
a
corresponding
mathematical
argument
in
which
†
I
and
‡
I
are
identified
[say,
via
the
natural
isomorphisms
of
†
I,
†
I
with
I],
i.e.,
in
which
“J”
is
replaced
by
“M”.
Indeed,
for
instance,
one
verifies
immediately,
just
as
in
the
situation
of
Exam-
ple
2.3.1,
that
the
fundamental
groups
of
J
and
M
are
not
isomorphic.
That
is
to
say,
just
as
in
the
situation
of
Example
2.3.1,
the
topological
space
J
is
struc-
turally/qualitatively
very
different
from
the
topological
space
M.
46
SHINICHI
MOCHIZUKI
§2.4.
Logical
AND
“∧”
vs.
logical
OR
“∨”
The
essential
mathematical
content
of
the
elementary
geometric
examples
dis-
cussed
in
§2.3
may
be
reformulated
in
terms
of
the
symbolic
logical
relators
AND
“∧”
and
OR
“∨”.
This
reformulation
renders
the
elementary
geometric
examples
of
§2.3
in
a
form
that
is
even
more
directly
technically
related
to
various
central
aspects
of
the
assertions
of
the
RCS
concerning
inter-universal
Teichmüller
theory.
Example
2.4.1:
“∧”
vs.
“∨”
for
adjacent
oriented
line
segments.
(i)
Recall
the
situation
discussed
in
Example
2.3.2.
Thus,
J
⊇
†
I
⊇
†
β
=
γ
J
=
‡
α
⊆
‡
I
⊆
J,
i.e.,
(AOL1)
the
following
condition
holds:
†
†
γ
J
=
β
⊆
I
∧
‡
‡
γ
J
=
α
⊆
I
.
On
the
other
hand,
if
one
identifies
†
I,
‡
I,
then
one
obtains
a
topological
space
∼
∼
M
→
L,
i.e.,
a
loop.
Here,
“
→
”
denotes
the
natural
isomorphism
discussed
in
Example
2.3.2,
(ii).
Now
suppose
that
we
are
given
a
connected
subspace
γ
I
⊆
I
whose
image
in
the
quotient
I
L
=
I/
α
∼
β
coincides
with
γ
L
⊆
L,
i.e.,
with
the
image
of
γ
J
⊆
J
via
the
composite
of
the
quotient
J
M
=
J/
†
I
∼
‡
I
with
∼
the
natural
isomorphism
M
→
L.
Then
observe
that
(AOL2)
the
following
condition
holds:
γ
I
∈
{α,
β},
i.e.,
γ
I
=
β
⊆
I
∨
γ
I
=
α
⊆
I
.
Of
course,
(AOL3)
the
essential
mathematical
content
discussed
in
this
condition
(AOL2)
may
be
formally
described
as
a
condition
involving
the
AND
relator
“∧”:
β
∈
{α,
β}
∧
α
∈
{α,
β}
.
But
the
essential
mathematical
content
of
the
OR
relator
“∨”
statement
in
(AOL2)
remains
unchanged.
(ii)
On
the
other
hand,
[unlike
the
case
with
γ
J
!]
(AOL4)
the
following
condition
does
not
hold:
γ
I
=
β
⊆
I
∧
γ
I
=
α
⊆
I
.
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
47
That
is
to
say,
in
summary,
the
operation
of
identifying
†
I,
‡
I
—
e.g.,
on
the
grounds
of
“redundancy”
[cf.
the
discussion
of
Example
2.3.2]
—
has
the
effect
of
passing
from
a
situation
in
which
the
AND
relator
“∧”
holds
[cf.
(AOL1)]
to
a
situation
in
which
the
OR
relator
“∨”
holds
[cf.
(AOL2),
(AOL3)],
but
the
AND
relator
“∧”
does
not
hold
[cf.
(AOL4)]!
(iii)
It
turns
out
that
this
phenomenon
—
i.e.,
of
an
identification
of
“redundant
copies”
leading
to
a
passage
from
the
validity
of
an
“∧”
relation
to
the
validity
of
an
“∨”
relation
coupled
with
the
invalidity
of
an
“∧”
relation
—
forms
a
very
precise
model
of
the
situation
that
arises
in
the
assertions
of
the
RCS
concerning
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
§3.2,
§3.4
below].
Example
2.4.2:
Differentials
on
oriented
line
segments.
(i)
In
the
situation
of
Example
2.4.1,
one
way
to
understand
the
gap
between
(AOL1)
and
(AOL4)
—
i.e.,
the
central
issue
of
whether
the
AND
relator
“∧”
holds
or
does
not
hold
—
is
to
think
in
terms
of
the
restriction
to
I
⊆
R
of
the
coordinate
function
“x”
of
Example
2.2.1.
Indeed,
(AOD1)
one
may
interpret
(AOL4)
as
the
statement
that
the
coordinate
functions
“x”
on
the
two
copies
†
I,
‡
I
that
constitute
J
do
not
glue
together
to
form
a
single,
well-defined
R-valued
function
on
J
[that
is
to
say,
since
it
is
not
clear
whether
the
value
of
such
a
function
on
γ
J
⊆
J
should
be
0
or
1,
i.e.,
such
a
function
is
not
well-defined
on
γ
J
⊆
J];
(AOD2)
on
the
other
hand,
(AOL1)
may
be
interpreted
as
the
statement
that
such
a
function
[i.e.,
obtained
by
gluing
together
the
coordinate
functions
“x”
on
the
two
copies
†
I,
‡
I
that
constitute
J]
can
indeed
be
defined
if
one
regards
its
values
as
being
[not
in
a
single
copy
of
R,
but
rather]
in
the
set
†,‡
R
obtained
by
gluing
together
two
distinct
copies
†
R,
‡
R
of
R
by
identifying
†
1
∈
†
R
with
‡
0
∈
‡
R.
(ii)
On
the
other
hand,
if,
instead
of
considering
the
coordinate
function
“x”,
one
considers
the
differential
“dx”
associated
to
this
coordinate
function
[cf.
the
discussion
of
Example
2.2.1],
then
one
observes
immediately
that
(AOD3)
the
differentials
“dx”
on
the
two
copies
†
I,
‡
I
that
constitute
J
do
indeed
glue
together
to
form
a
single,
well-defined
differential
on
J
that,
moreover,
is
compatible
with
the
quotient
J
M
=
J/
†
I
∼
‡
I
in
the
sense
that,
as
is
easily
verified,
it
arises
as
the
pull-back,
via
this
quotient
map
J
M,
of
a
[smooth]
differential
on
the
[smooth
manifold
constituted
by
the]
loop
M.
Note,
moreover,
that
the
gluings
and
compatibility
of
(AOD3)
may
be
achieved
without
considering
functions
or
differentials
valued
in
some
sort
of
complicated
[i.e.,
by
comparison
to
R!]
set
such
as
†,‡
R.
48
SHINICHI
MOCHIZUKI
(iii)
It
turns
out
[cf.
the
discussion
of
Example
2.4.1,
(iii)]
that
the
phenomenon
discussed
in
(AOD3)
is
closely
related
to
the
situation
that
arises
in
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
§3.2
below].
Example
2.4.3:
Representation
via
subgroup
indices
of
“∧”
vs.
“∨”.
(i)
Let
A
be
an
abelian
group
and
B
1
,
B
2
⊆
A
subgroups
of
A
such
that
B
1
∩
B
2
has
finite
index
in
B
1
and
B
2
.
Then
one
may
define
a
positive
rational
number,
which
we
call
the
index
of
B
2
relative
to
B
1
,
def
[B
1
:
B
2
]
=
[B
1
:
B
1
∩
B
2
]/[B
2
:
B
1
∩
B
2
]
∈
Q
>0
.
Thus,
[B
1
:
B
2
]
·
[B
2
:
B
1
]
=
1;
when
B
2
⊆
B
1
,
this
notion
of
index
coincides
with
the
usual
notion
of
the
index
of
B
2
in
B
1
.
(ii)
Let
n
be
a
positive
integer
≥
2.
Consider
the
diagram
of
group
homomor-
phisms
n·
n·
G
1
−→
G
2
−→
G
3
—
where,
for
i
=
1,
2,
3,
G
i
denotes
a
copy
of
[the
additive
group
of
rational
integers]
Z,
and
the
arrows
are
given
by
multiplication
by
n.
For
i
=
1,
2,
3,
write
G
Q
i
=
G
i
⊗
Z
Q
for
the
tensor
product
of
G
i
over
Z
with
Q.
Then
observe
that
this
diagram
induces
a
diagram
of
group
isomorphisms
def
G
Q
1
∼
→
G
Q
2
∼
→
G
Q
3
—
i.e.,
in
which
the
arrows
are
isomorphisms.
Let
us
use
these
isomorphisms
to
Q
identify
the
groups
G
Q
i
,
for
i
=
1,
2,
3,
and
denote
the
resulting
group
by
G
∗
.
(iii)
Observe
that
the
first
diagram
of
(ii)
is
structurally
reminiscent
of
the
object
J
discussed
in
Examples
2.3.2,
2.4.1,
2.4.2,
i.e.,
if
one
regards
·
the
first
arrow
of
the
first
diagram
of
(ii)
as
corresponding
to
†
I,
·
the
second
arrow
of
the
first
diagram
of
(ii)
as
corresponding
to
‡
I,
and
·
G
1
,
G
2
,
and
G
3
as
corresponding
to
†
α,
†
β
=
‡
α,
and
‡
β,
respectively.
Here,
we
observe
that
G
2
appears
simultaneously
as
the
codomain
of
the
arrow
n·
n·
G
1
−→
G
2
AND
[cf.
(AOL1)!]
as
the
domain
of
the
arrow
G
2
−→
G
3
.
Moreover,
we
may
consider
indices
of
G
1
,
G
2
,
and
G
3
as
subgroups
of
G
Q
∗
[G
2
:
G
1
]
=
[G
1
:
G
2
]
−1
=
n;
[G
3
:
G
2
]
=
[G
2
:
G
3
]
−1
=
n;
[G
3
:
G
1
]
=
[G
1
:
G
3
]
−1
=
n
2
in
a
consistent
fashion,
i.e.,
in
a
fashion
that
does
not
give
rise
to
any
contra-
dictions.
(iv)
On
the
other
hand,
suppose
that
we
delete
the
“distinct
labels”
G
1
,
G
2
,
G
3
from
the
copies
of
Z
considered
in
the
first
diagram
of
(ii).
This
yields
a
diagram
Z
n·
−→
Z
n·
−→
Z
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
49
in
which
the
second
arrow
may
be
regarded
as
a
copy
of
the
first
arrow.
This
situation
might
motivate
some
observers
to
conclude
that
these
two
arrows
are
“redundant”
and
hence
should
be
identified
with
one
another
—
cf.
the
discus-
sion
of
the
quotient
J
M
in
Example
2.3.2,
(ii)
—
to
form
a
diagram
n·
Z
consisting
of
a
single
copy
of
Z
and
the
endomorphism
of
this
single
copy
of
Z
given
by
multiplication
by
n.
At
first
glance,
this
operation
of
identification
may
appear
to
give
rise
to
various
“contradictions”
in
the
computation
of
the
index,
i.e.,
such
as
1
=
[G
1
:
G
1
]
=
[Z
:
Z]
=
[G
2
:
G
1
]
=
n
≥
2
and
so
on.
In
fact,
however,
if
one
takes
into
account
the
OR
relator
“∨”
[but
not
the
AND
relator
“∧”!]
relations
that
one
obtains
upon
executing
the
identification
operation
in
question
[cf.
(AOL2),
(AOL4)!],
then
one
concludes
that
[after
exe-
cuting
the
identification
operation
in
question!]
each
of
the
indices
[G
i
:
G
j
],
for
i,
j
∈
{1,
2,
3},
may
only
be
computed
up
to
multiplication
by
an
integral
power
of
n,
i.e.,
that
each
index
[G
i
:
G
j
],
for
i,
j
∈
{1,
2,
3},
is
only
well-defined
as
“some
def
indeterminate
element”
of
n
Z
=
{n
m
|
m
∈
Z}
⊆
Q
>0
.
In
particular,
in
fact
there
is
no
contradiction.
Example
2.4.4:
Logical
“∧/∨”
vs.
“narrative
∧/∨”.
Consider
the
following
argument
concerning
a
natural
number
x
∈
{1,
3},
i.e.,
a
natural
number
for
which
it
holds
that
(x
=
1)
∨
(x
=
3):
(Nar1)
Suppose
that
x
=
3.
Then
it
follows
that
x
=
3
>
2.
That
is
to
say,
we
conclude
that
x
>
2.
(Nar2)
Since
(x
=
1)
∨
(x
=
3),
we
may
consider
the
case
x
=
1.
Then,
by
applying
the
conclusion
of
(Nar1),
we
conclude
that
1
=
x
>
2,
i.e.,
that
1
>
2
—
a
contradiction!
Of
course,
this
argument
is
completely
fallacious!
On
the
other
hand,
it
yields
a
readily
understood
concrete
example
of
the
absurdity
that
arises
when,
as
is
in
effect
done
in
(Nar2),
logical
OR
“∨”
is
confused
with
logical
AND
“∧”!
In
various
contexts,
this
sort
of
confusion
can
arise
from
the
ambiguity
of
various
narrative
expressions
that
appear
in
the
discussion
of
a
mathematical
argument.
This
sort
of
ambiguity
can
lead
to
a
situation
in
which
a
“narrative
AND
∧”
—
i.e.,
the
fact
that
in
a
particular
narrative
exposition
of
an
argument,
one
performs
both
the
task
of
considering
the
case
“x
=
3”
[cf.
(Nar1)]
and
the
task
of
considering
the
case
“x
=
1”
[cf.
(Nar2)]
—
is
mistakenly
construed
as
a
logical
AND
“∧”.
In
a
similar
vein,
one
may
consider
situations
in
which
the
roles
played
by
“∧”
and
“∨”
are
reversed,
i.e.,
in
which
a
“narrative
OR
∨”
—
i.e.,
the
fact
that
in
a
particular
narrative
exposition
of
an
argument,
one’s
attention
is
concentrated
50
SHINICHI
MOCHIZUKI
either
on
the
task
of
considering
one
situation
or
on
the
task
of
considering
another
situation
—
is
mistakenly
construed
as
a
logical
OR
“∨”.
Indeed,
it
appears
that
one
fundamental
cause,
in
the
context
of
the
essential
logical
structure
of
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
Example
2.4.5
below!],
of
the
confusion
on
the
part
of
some
mathematicians
between
logical
AND
“∧”
and
logical
OR
“∨”
lies
precisely
in
this
sort
of
confusion
between
“narrative
∧/∨”
and
logical
“∧/∨”.
Example
2.4.5:
Numerical
representation
of
“∧”
vs.
“∨”.
(i)
A
slightly
more
sophisticated
numerical
representation
of
the
difference
between
“∧”
and
“∨”
—
which
in
fact
mirrors
the
essential
logical
structure
of
inter-universal
Teichmüller
theory
in
a
very
direct
fashion
—
may
be
given
as
follows
[cf.
[Alien],
Example
3.11.4].
Indeed,
the
essential
logical
flow
of
inter-
universal
Teichmüller
theory
may
be
summarized
as
follows:
·
one
starts
with
the
definition
of
an
object
called
the
Θ-link;
·
one
then
constructs
a
complicated
apparatus
that
is
referred
to
as
the
multiradial
representation
of
the
Θ-pilot
[cf.
[IUTchIII],
Theorem
3.11];
·
finally,
one
derives
a
final
numerical
estimate
[cf.
[IUTchIII],
Corol-
lary
3.12]
in
an
essentially
straightforward
fashion
from
the
multiradial
representation
of
the
Θ-pilot.
(ii)
An
elementary
model
of
this
essential
logical
flow
may
be
given
by
means
of
real
numbers
A,
B
∈
R
>0
and
,
N
∈
R
such
that
0
≤
≤
1
in
the
following
way:
·
Θ-link:
def
N
=
−2B
∧
N
=
−A
;
def
·
multiradial
representation
of
the
Θ-pilot:
N
=
−2A
+
∧
N
=
−A
;
·
final
numerical
estimate:
−2A
+
=
−A,
hence
A
=
,
i.e.,
A
≤
1.
Thus,
the
definition
of
the
Θ-link
and
the
construction
of
the
multiradial
represen-
tation
of
the
Θ-pilot
are
meaningful/nontrivial
precisely
on
account
of
the
validity
of
the
AND
relator
“∧”,
which
is
rendered
possible,
in
the
definition
of
the
Θ-link,
precisely
by
allowing
the
real
numbers
A,
B
to
be
[a
priori]
distinct
real
numbers
—
cf.
(AOL1)
vs.
(AOL4),
where
we
think
in
terms
of
the
correspondences
B
−2B
†
←→
←→
†
I,
N
←→
γ
J
,
A
β,
−A
←→
‡
−2A
α,
←→
‡
←→
I
‡
β.
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
51
The
passage
from
the
multiradial
representation
of
the
Θ-pilot
to
the
final
numerical
estimate
is
then
immediate/straightforward/logically
transparent.
(iii)
By
contrast,
if,
in
the
elementary
numerical
model
of
(ii),
one
replaces
“∧”
by
“∨”,
then
our
elementary
numerical
model
of
the
logical
structure
of
inter-
universal
Teichmüller
theory
takes
the
following
form:
·
“∨”
version
of
Θ-link:
def
def
∨
N
=
−A
N
=
−2B
def
[cf.
N
=
−2A
∨
N
=
−A
];
def
·
“∨”
version
of
multiradial
representation
of
the
Θ-pilot:
N
=
−2A
+
∨
N
=
−A
;
·
final
numerical
estimate:
−2A
+
=
−A,
hence
A
=
,
i.e.,
A
≤
1.
That
is
to
say,
the
use
of
distinct
real
numbers
A,
B
in
the
definition
of
the
“∨”
ver-
sion
of
Θ-link
seems
entirely
superfluous
[cf.
(AOL2),
relative
to
the
correspon-
dences
discussed
in
(ii)].
This
motivates
one
to
identify
A
and
B
—
i.e.,
to
suppose
“for
the
sake
of
simplicity”
that
A
=
B
—
which
then
has
the
effect
of
rendering
the
definition
of
the
original
“∧”
version
of
the
Θ-link
invalid/self-contradictory
[cf.
(AOL4),
relative
to
the
correspondences
discussed
in
(ii)].
Once
one
identifies
A
and
B,
i.e.,
once
one
supposes
“for
the
sake
of
simplicity”
that
A
=
B,
the
pas-
sage
from
the
“∨”
version
of
Θ-link
to
the
resulting
“∨”
version
of
the
multiradial
representation
of
the
Θ-pilot
then
seems
entirely
meaningless/devoid
of
any
interesting
content.
The
passage
from
the
resulting
meaningless
“∨”
version
of
the
multiradial
representation
of
the
Θ-pilot
to
the
final
numerical
estimate
then
seems
abrupt/mysterious/entirely
unjustified,
i.e.,
put
another
way,
looks
as
if
one
erroneously
replaced
the
“∨”
in
the
meaningless
“∨”
version
of
the
multiradial
representation
of
the
Θ-pilot
by
an
“∧”
without
any
mathematical
justification
whatsoever.
It
is
precisely
this
pernicious
chain
of
misunderstandings
emanating
from
the
“redundancy”
assertions
of
the
RCS
that
has
given
rise
to
a
substantial
amount
of
unnecessary
confusion
concerning
inter-universal
Teichmüller
theory.
(iv)
Before
proceeding,
we
observe
that
the
sort
of
confusion
discussed
in
(iii)
between
“∧”
and
“∨”
can
occur
as
the
result
of
any
of
the
following
phenomena:
(AOC1)
a
confusion
between
“narrative
∧/∨”
and
logical
“∧/∨”,
as
discussed
in
Example
2.4.4;
(AOC2)
thinking
in
terms
of
the
“fake
∧”
of
(AOL3),
i.e.,
which,
though
for-
mulated
as
a
logical
AND
“∧”
relation,
is
in
fact,
substantively
speaking,
a
logical
OR
“∨”
relation;
52
SHINICHI
MOCHIZUKI
(AOC3)
the
symptom
(Syp2)
discussed
in
§3.6
below,
i.e.,
a
desire
to
see
the
“proof”
of
some
sort
of
commutative
diagram
or
“compatibility
prop-
erty”
to
the
effect
that
taking
log-volumes
of
pilot
objects
in
the
domain
and
codomain
of
the
Θ-link
yields
the
same
real
number;
(AOC4)
a
fundamental
misunderstanding
—
which
is
often
closely
inter-
twined
with
the
symptom
(Syp2)
discussed
in
(AOC3)
—
of
the
meaning
of
the
crucial
closed
loop
of
§3.10,
(Stp7),
(Stp8),
below
[cf.
§3.10,
(Stp7),
(Stp8),
as
well
as
the
following
discussion].
(v)
Let
us
refer
to
the
“∧”
version
of
inter-universal
Teichmüller
theory
dis-
cussed
in
(ii)
—
i.e.,
the
original
version
of
inter-universal
Teichmüller
theory,
in
which
one
interprets
the
Θ-link
as
a
logical
AND
“∧”
relation
—
as
AND-IUT.
Thus,
AND-IUT
=
IUT
is
the
original
version
of
inter-universal
Teichmüller
theory.
Let
us
refer
to
the
“∨”
version
of
inter-universal
Teichmüller
theory
discussed
in
(iii)
—
i.e.,
the
version
of
inter-universal
Teichmüller
theory
that
arises
if
one
[mis-
takenly!]
interprets
the
Θ-link
as
a
logical
OR
“∨”
relation
—
as
OR-IUT.
As
discussed
in
(iii),
in
OR-IUT,
one
is
motivated
to
implement
the
RCS-identifications
of
RCS-redundant
copies
of
objects
—
i.e.,
in
the
language
of
(iii),
to
“identify
A
and
B”
—
and
hence
to
conclude
that
OR-IUT
=⇒
RCS-IUT,
where
we
recall
that
“RCS-IUT”
refers
to
the
version
of
inter-universal
Teichmüller
theory
obtained
by
implementing
the
RCS-identifications
of
RCS-redundant
copies
of
objects
[cf.
the
discussion
of
§1.2].
On
the
other
hand,
it
is
not
difficult
to
see
that
in
RCS-IUT,
one
is
forced
to
work
with
a
(NeuORInd)
indeterminacy
[cf.
the
discussion
at
the
end
of
§3.4
below,
as
well
as
the
discussion
of
(ΘORInd)
in
§3.11
below],
i.e.,
to
interpret
˙
relation
[that
is
to
say,
a
logical
OR
“∨”
the
Θ-link
as
a
logical
XOR
“
∨”
relation
such
that
the
corresponding
logical
AND
“∧”
relation
cannot
hold
—
cf.
the
discussion
of
(iii)].
In
particular,
we
conclude
that
RCS-IUT
=⇒
XOR-IUT
=⇒
OR-IUT
[where
the
second
“
=⇒
”
is
a
consequence
of
well-known
general
properties
of
Boolean
operators],
i.e.,
in
summary:
(XOR/RCS)
we
have
equivalences
XOR-IUT
⇐⇒
OR-IUT
⇐⇒
RCS-IUT.
In
the
following,
I
shall
refer
to
the
school
of
thought
[i.e.,
in
the
sense
of
a
“collection
of
closely
interrelated
ideas”]
surrounding
OR-IUT
as
ORS,
i.e.,
the
“OR
school
[of
thought]”,
and
to
the
school
of
thought
surrounding
XOR-IUT
as
XORS,
i.e.,
the
“XOR
school
[of
thought]”.
Thus,
XORS
=
ORS
=
RCS.
(vi)
On
the
other
hand,
one
may
also
consider
yet
another
version
of
inter-
universal
Teichmüller
theory,
also
motivated
by
the
discussion
of
(iii),
which
we
refer
to
as
EssOR-IUT,
i.e.,
“essentially
OR
IUT”.
This
is
the
version
of
inter-
universal
Teichmüller
theory
in
which
one
accepts,
at
the
level
of
formal
definitions,
the
logical
AND
“∧”
version
of
the
Θ-link
as
in
(ii),
i.e.,
without
identifying
A
and
B,
but
[for
some
unexplained
reason!]
one
then
arbitrarily
shifts,
when
considering
the
multiradial
representation
of
the
Θ-pilot,
to
the
logical
OR
“∨”
interpretation
of
the
multiradial
representation
of
the
Θ-pilot,
i.e.,
as
in
(iii).
That
is
to
say,
as
the
name
“EssOR-IUT”
suggests,
the
fundamental
logical
AND
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
53
“∧”
property
of
the
Θ-link
is
never
actually
used
in
any
sort
of
meaningful
way
in
EssOR-IUT.
In
particular,
(EssOR/RCS)
although,
at
a
purely
formal
level,
EssOR-IUT
rejects
RCS-IUT,
the
essential
logical
structure
of
EssOR-IUT
still
nevertheless
gives
rise
to
the
abrupt/mysterious/entirely
unjustified
transition
discussed
in
(iii)
to
the
final
numerical
estimate.
It
appears
that
the
“arbitrarily
shift”
referred
to
above
is
often
precipitated
by
the
various
phenomena
discussed
in
(iv)
[cf.,
especially,
(AOC3),
(AOC4)].
In
the
following,
I
shall
refer
to
the
school
of
thought
[i.e.,
in
the
sense
of
a
“collection
of
closely
interrelated
ideas”]
surrounding
EssOR-IUT
as
EssORS,
i.e.,
the
“essen-
tially
OR
school
[of
thought]”.
(vii)
In
general,
at
the
level
of
formalities
of
Boolean
operators,
“∧
⇒
∨”,
but
“∨
⇒
∧”.
In
particular,
in
the
context
of
the
transition
to
the
final
numerical
estimate
of
inter-universal
Teichmüller
theory,
(∨
⇒
∧)
it
appears
entirely
hopeless/unrealistic
to
pass
from
the
“∨”
version
of
the
multiradial
representation
of
the
Θ-pilot
to
the
“∧”
version
of
the
multiradial
representation
of
the
Θ-pilot.
This
is
precisely
the
“abrupt/mysterious/entirely
unjustified”
transition
[to
the
final
numerical
estimate]
discussed
in
(iii).
(viii)
The
discussion
of
(v),
(vi),
(vii)
may
be
summarized
as
follows
[cf.
also
the
discussion
of
§1.2,
§1.3]:
·
The
fundamental
misunderstanding
on
the
part
of
adherents
of
the
RCS
=
ORS
=
XORS
to
the
effect
that
OR-IUT
or
XOR-IUT
is
indeed
the
content
of
AND-IUT
=
IUT
leads
to
the
mistaken
interpretation
of
the
assertion
(XOR/RCS)
as
an
equivalence
between
AND-IUT
=
IUT
and
RCS-IUT.
·
The
fundamental
misunderstanding
on
the
part
of
adherents
of
the
RCS
=
ORS
=
XORS
to
the
effect
that
OR-IUT
or
XOR-IUT
is
indeed
the
content
of
AND-IUT
=
IUT
leads
to
the
mistaken
interpretation
of
the
assertion
(∨
⇒
∧)
as
a
logical
flaw
in
AND-IUT
=
IUT.
·
The
fundamental
misunderstanding
on
the
part
of
adherents
of
the
EssORS
to
the
effect
that
EssOR-IUT
is
indeed
the
content
of
AND-IUT
=
IUT
leads
either
to
the
mistaken
interpretation
of
the
assertion
(∨
⇒
∧)
as
a
logical
flaw
in
AND-IUT
=
IUT
or
to
the
mistaken
interpretation
of
the
assertion
(∨
⇒
∧)
as
an
indication
the
existence
of
some
sort
of
infinitely
complicated
and
mysterious
argument
—
i.e.,
for
concluding
that
“∨
⇒
∧”!
—
in
inter-universal
Teichmüller
theory
that
requires
years
of
concerted
effort
to
understand.
Thus,
as
the
descriptive
“essential”
suggests,
there
is
in
fact,
from
the
point
of
view
of
the
essential
logical
structure
under
consideration,
very
little
difference
between
EssORS
and
XORS
=
ORS
=
RCS
or
between
EssOR-IUT
and
XOR-IUT
⇐⇒
OR-IUT
⇐⇒
RCS-IUT.
·
In
fact,
the
correct
interpretation
of
the
assertion
(∨
⇒
∧)
consists
of
the
conclusion
that
neither
XOR-IUT
⇐⇒
OR-IUT
⇐⇒
RCS-IUT
nor
54
SHINICHI
MOCHIZUKI
EssOR-IUT
has
any
direct
logical
relationship
to
AND-IUT
=
IUT.
Assertions
of
various
schools
of
thought
Actual
mathematical
content
RCS
=
ORS
=
XORS:
“IUT
⇔
RCS-IUT”
“XOR-IUT
⇔
OR-IUT
⇔
RCS-IUT”
RCS
=
ORS
=
XORS:
“IUT
is
logically
flawed.”
EssORS:
either
“IUT
is
logically
flawed.”
or
“The
logical
structure
of
IUT
is
infinitely
complicated/mysterious.”
“∨
⇒
∧”,
which
implies
that
“(AND-IUT
=)
IUT
⇒
RCS-IUT”,
“(AND-IUT
=)
IUT
⇒
OR-IUT”,
“(AND-IUT
=)
IUT
⇒
XOR-IUT”
“∨
⇒
∧”,
which
implies
that
“(AND-IUT
=)
IUT
⇒
EssOR-IUT”
Here,
we
observe
that
the
above
analysis
is
in
some
sense
remarkable
in
that
it
makes
explicit
the
fact
that,
if
one
forgets
the
arbitrary
label
“inter-universal
Te-
ichmüller
theory”
placed
on
XOR-IUT
or
OR-IUT
or
EssOR-IUT
by
adherents
of
the
RCS
=
ORS
=
XORS
or
the
EssORS,
then
[perhaps
somewhat
surprisingly!]
(MthVl)
there
is
in
fact
no
disagreement
among
any
of
the
parties
involved
with
regard
to
the
mathematical
validity
of
the
mathematical
assertions
(XOR/RCS)
and
(∨
⇒
∧).
Indeed,
this
state
of
affairs
may
be
understood
as
in
some
sense
highlighting
the
es-
sentially
social/political/psychological,
i.e.,
in
summary,
non-mathematical
nature
of
the
entirely
unnecessary
confusion
that
has
arisen
concerning
inter-universal
Te-
ichmüller
theory
[cf.
the
discussion
of
§1.8,
§1.11].
The
above
observations
are
summarized
in
the
“dictionary
of
assertions”
given
above.
Example
2.4.6:
Carry
operations
in
arithmetic,
geometry,
and
Boolean
logic.
(i)
Observe
that
if,
in
the
situation
of
Example
2.4.3,
(ii),
one
focuses
one’s
attention
on
the
subset
D
4−i
⊆
G
i
in
the
copy
of
Z
denoted
by
G
i
,
where
i
=
1,
2,
3,
corresponding
to
{0,
1,
2,
.
.
.
,
n
−
1}
⊆
Z,
then
the
situation
considered
in
Example
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
55
2.4.3,
(ii),
closely
resembles
the
situation
that
arises
in
elementary
arithmetic
com-
putations
—
such
as
addition
and
multiplication
—
involving
base
n
expansions
of
natural
numbers.
That
is
to
say,
one
may
think
of
·
D
1
as
the
first
digit,
i.e.,
when
n
=
10,
the
“ones
digit”,
·
D
2
as
the
second
digit,
i.e.,
when
n
=
10,
the
“tens
digit”,
and
·
D
3
as
the
third
digit,
i.e.,
when
n
=
10,
the
“hundreds
digit”
of
such
an
expansion.
When
performing
such
elementary
arithmetic
computations
—
such
as
addition
and
multiplication
—
involving
base
n
expansions
of
natural
numbers,
recall
that
it
is
of
fundamental
importance
to
take
into
account
the
various
carry
operations
that
occur.
In
particular,
we
observe
that
the
use
of
distinct
labels
for
distinct
digits
plays
a
fundamental
role
in
elementary
arithmetic
computations
involving
base
n
expansions
of
natural
numbers
—
cf.
the
distinct
labels
“G
1
”,
“G
2
”,
“G
3
”
in
the
discussion
of
Example
2.4.3,
(ii),
(iii).
This
situation
is
reminiscent
of
the
important
role
played
by
the
distinct
labels
“A”,
“B”
in
the
“Θ-link”
of
Example
2.4.5,
(ii).
Note,
morever,
that
deletion/confusion
of
these
distinct
labels
for
distinct
digits
has
the
effect
of
completely
invalidating,
at
least
in
the
usual
“strict
sense”,
elementary
arithmetic
computations
involving
base
n
expansions
of
natural
numbers
—
cf.
the
situation
considered
in
Example
2.4.3,
(iv).
On
the
other
hand,
if
one
restricts
one’s
attention
to
a
specific
computational
algorithm
[involving,
say,
addition
and
multiplication
operations],
then
in
fact
it
is
often
possible
—
i.e.,
de-
pending
on
the
content
of
the
specific
computational
algorithm
under
consideration
—
to
obtain
estimates
to
the
effect
that
applying
the
algorithm
either
with
or
without
the
use
of
distinct
labels
for
distinct
digits
in
the
base
n
expansions
of
the
natural
numbers
involved
in
fact
yields
the
same
result,
up
to
some
explicitly
bounded
discrepancy.
[For
instance,
when
n
=
10,
any
algorithm
that
only
involves
addition
and
multipli-
cation
operations
yields
the
same
result
modulo
9
(=
10
−
1),
regardless
of
whether
or
not
one
uses
distinct
labels
for
distinct
digits
in
decimal
expansions
of
natural
numbers.]
Such
estimates
are
reminiscent
of
the
“multiradial
representation”
of
Example
2.4.5,
(ii).
(ii)
Observe
that
the
discussion
of
·
adjacent
oriented
line
segments
“
†
I”,
“
‡
I”
and
·
oriented
loops
“L”,
“M”
in
Examples
2.3.1,
2.3.2,
2.4.1,
2.4.2
[cf.
also
the
discussion
of
Examples
2.4.3,
(iii);
2.4.5,
(ii)]
may
be
regarded
as
a
sort
of
limiting
case
of
the
discussion
of
base
n
expansions
of
natural
numbers
in
(i)
above,
i.e.,
if
one
·
considers
the
real
numbers
obtained
by
dividing
the
natural
numbers
≤
2n
in
the
discussion
of
(i)
above
by
n
and
then
56
SHINICHI
MOCHIZUKI
·
passes
to
the
limit
n
→
+∞.
That
is
to
say,
in
summary,
the
adjacency
of
the
oriented
line
segments
“
†
I”,
“
‡
I”
may
be
under-
stood
as
a
sort
of
continuous,
geometric
representation
of
the
carry
operation
that
appears
in
elementary
arithmetic
computations
involving
base
n
expansions
of
natural
numbers.
(iii)
From
the
point
of
view
of
discussions
of
the
logical
structure
of
math-
ematical
arguments
represented
in
terms
of
Boolean
operators
such
as
logical
AND
“∧”
and
logical
OR
“∨”,
it
is
of
interest
to
consider
the
discussion
of
(i)
above
in
the
binary
case,
i.e.,
the
case
n
=
2.
We
begin
our
discussion
of
the
binary
case
by
recalling
the
following
well-known
facts:
·
multiplication
in
the
field
F
2
=
{0,
1}
may
be
regarded
as
corresponding
to
the
Boolean
operator
AND
“∧”;
·
addition
in
the
field
F
2
=
{0,
1}
may
be
regarded
as
corresponding
to
the
Boolean
operator
XOR
[i.e.,
“exclusive
OR”],
which
we
denote
by
˙
“
∨”;
·
“carry-addition”
in
the
truncated
ring
of
Witt
vectors
F
2
×
F
2
—
∼
i.e.,
addition
of
two
elements
of
F
2
→
{0}
×
F
2
⊆
F
2
×
F
2
,
regarded
as
Teichmüller
representatives
in
the
truncated
ring
of
Witt
vectors
Z/4Z,
that
is
to
say,
an
addition
operation
in
which
one
allows
for
the
carry
operation
[cf.
the
discussion
of
(i)!]
to
the
first
factor
of
F
2
×
F
2
—
may
be
regarded
as
corresponding
to
an
operator
that
we
shall
refer
to
as
the
¨
”;
thus,
we
have
“COR”,
or
“carry-OR”,
operator
and
denote
by
“
∨
¨
=
(∧,
∨)
˙
∨
¨
0
=
(0,
0);
1
∨
¨
0
=
0
∨
¨
1
=
(0,
1);
1
∨
¨
1
=
(1,
0)].
[so
0
∨
These
well-known
facts
case
may
be
summarized
as
follows:
¨
=
∧
∨)
˙
Conventional
mixed-characteristic/“carry”
addition
in
Z
consid-
(
∨
¨
”
—
may
be
described
in
terms
of
the
“splitting”
ered
modulo
4
—
i.e.,
“
∨
of
the
natural
surjection
Z
F
2
determined
by
the
“Teichmüller
repre-
sentatives”
0,
1
∈
Z
via
the
equation
¨
=
(∧,
∨)
˙
∨
¨
”
as
an
operation
obtained
by
“stacking”
mul-
—
i.e.,
which
exhibits
“
∨
˙
in
F
2
.
Here,
we
note
tiplication
“∧”
in
F
2
on
top
of
addition
“
∨”
that
this
splitting
via
Teichmüller
representatives
0,
1
∈
Z
is
compatible
with
the
multiplicative
structures
in
Z
and
F
2
,
but
not
with
the
addi-
tive
structures
in
Z
and
F
2
.
Put
another
way,
one
may
think
of
the
ring
structures
of
Z
and
F
2
as
structures
that
share
a
common
multiplicative
structure
[cf.
“∧”!],
but
do
not
share
a
common
additive
structure
[cf.
˙
“
∨”!].
These
observations
will
be
of
fundamental
importance
in
the
theory
developed
in
§3
[cf.,
especially,
the
discussion
at
the
beginning
of
§3.10].
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
57
(iv)
Some
readers
may
object
to
the
comparisons
and
analogies
between
inter-
universal
Teichmüller
theory
and
the
“mathematics
of
carry
operations”
dis-
cussed
in
(i),
(ii),
and
(iii)
as
being
inappropriate
on
the
grounds
that
this
mathe-
matics
of
carry
operations
is
much
too
“trivial”
to
be
of
any
substantive
interest.
On
the
other
hand,
we
observe
that
the
mathematics
of
carry
operations
discussed
in
(i),
(ii),
and
(iii)
is
intimately
intertwined
with
numerous
important
develop-
ments
in
the
history
of
mathematics.
With
regard
to
the
content
of
(i),
we
recall
that
the
use
of
place-value
decimal
numerals,
i.e.,
that
make
use
of
notation
for
zero,
appears
to
date
back
to
Indian
texts
and
inscriptions
from
the
7-th
to
9-th
centuries
AD.
Such
numerals
also
reached
the
Arab
world
during
this
period,
but
this
Hindu-Arabic
numeral
system
apparently
only
became
widely
used
in
Europe
during
the
late
middle
ages,
between
the
13-th
and
15-th
centuries.
In
this
context,
one
noteworthy
development
was
the
book
Liber
Abaci
published
by
the
Italian
mathematician
Fibonacci
in
1202,
which
promoted
the
use
of
the
Hindu-
Arabic
numeral
system
in
Europe.
Here,
it
is
useful
to
recall
that,
by
comparison
to
earlier
numeral
systems,
such
as
the
Greco-Roman
and
Babylonian
systems,
place-
value
decimal
numerals
not
only
facilitate
elementary
arithmetic
computations
—
i.e.,
via
the
systematic
use
of
carry
operations,
as
discussed
in
(i)!
—
but
also
make
it
possible
to
express
all
—
hence,
in
particular,
infinitely
many
—
natural
numbers
by
means
of
finitely
many
symbols
—
i.e.,
unlike
earlier
numerical
systems,
in
which
only
finitely
many
natural
num-
bers
could
be
expressed
using
finitely
many
symbols.
This
revolutionary
importance
of
the
development
of
place-value
decimal
numerals
in
India
was
recognized,
for
in-
stance,
in
writings
of
the
18-th
century
French
mathematician
Laplace.
In
this
context,
it
is
also
of
interest
to
observe
that
the
discussion
in
(ii)
of
the
interpreta-
tion
of
the
discussion
of
(i)
in
terms
of
line
segments
is
reminiscent
of
the
discussion
of
the
“Euclidean
algorithm”
in
Euclid’s
Elements,
in
which
numbers
are
often
rep-
resented
as
lengths
of
line
segments.
Finally,
we
recall
that
the
Boolean
aspects
discussed
in
(iii)
played
an
important
role
in
the
[well-known!]
development
of
modern
digital
computers
in
the
20-th
century.
The
gluings
of
adjacent
line
segments
discussed
in
Examples
2.3.2,
2.4.1,
2.4.2
may
in
some
sense
be
regarded
as
a
sort
of
optimized
elementary
geometric/combinatorial
representation
of
the
essential
logical
“∧/∨”
structure
surrounding
a
gluing
in
a
fashion
that
is
qualitatively
entirely
structually
similar
to
the
gluings
that
occur
in
inter-universal
Teichmüller
theory,
which
will
be
discussed
in
more
detail
in
§3
below.
The
somewhat
more
numerical/arithmetic
situations
discussed
in
Examples
2.4.3,
2.4.5,
2.4.6
may
also
be
regarded
as
entirely
elementary
representations
of
this
essential
logical
“∧/∨”
structure
surrounding
a
gluing.
On
the
other
hand,
the
glu-
ing
operation
that
occurs
in
the
standard
construction
of
the
projective
line,
while
somewhat
less
elementary
than
the
previously
mentioned
examples,
also
constitutes
an
important
—
and,
moreover,
still
relatively
elementary!
—
representation
of
this
essential
logical
“∧/∨”
structure
surrounding
a
gluing.
Moreover,
this
example
of
the
projective
line
discussed
in
Example
2.4.7
is
more
directly
related
to
scheme-
theoretic
arithmetic
geometry
than
the
previously
mentioned
examples
and
58
SHINICHI
MOCHIZUKI
helps
to
motivate
the
subsequent
ring-/monoid-theoretic
Example
2.4.8,
which
may
literally
be
regarded,
i.e.,
in
a
much
more
rigorous,
technical
sense,
as
a
sort
of
miniature
qualitative
model
—
that
is
to
say,
so
to
speak,
a
sort
of
“preview”
—
of
the
gluing
constituted
by
the
Θ-link
of
inter-universal
Teichmüller
theory.
Example
2.4.7:
The
projective
line
as
a
gluing
of
ring
schemes
along
a
multiplicative
group
scheme.
In
the
following
discussion,
we
take
k
to
be
a
field
def
and
q
∈
k
to
be
an
element
such
that
q
3
=
q
[i.e.,
q
∈
{0,
1,
−1}].
Write
k
×
=
k\{0},
A
1
for
the
affine
line
Spec(k[T
])
over
k,
G
m
for
the
open
subscheme
Spec(k[T,
T
−1
])
of
A
1
obtained
by
removing
the
origin.
Thus,
the
standard
coordinate
T
on
A
1
,
∼
∼
G
m
determines
natural
bijections
A
1
(k)
→
k,
G
m
(k)
→
k
×
of
the
respective
sets
of
k-rational
points
of
A
1
,
G
m
with
corresponding
subsets
of
k.
Also,
we
recall
that
A
1
is
equipped
with
a
well-known
natural
structure
of
ring
scheme
over
k,
while
G
m
is
equipped
with
a
well-known
natural
structure
of
[multiplicative]
group
scheme
over
k.
(i)
Write
†
A
1
,
‡
A
1
for
the
k-ring
schemes
given
by
copies
of
A
1
equipped
with
the
respective
labels
“†”,
“‡”.
We
regard
†
A
1
as
being
further
equipped
with
the
∼
k-rational
point
†
q
−1
∈
†
A
1
(k)
(
→
k)
corresponding
to
the
multiplicative
inverse
of
the
element
q
∈
k
and
‡
A
1
as
being
further
equipped
with
the
k-rational
point
∼
‡
q
∈
‡
A
1
(k)
(
→
k)
corresponding
to
the
element
q
∈
k.
Similarly,
we
write
†
G
m
,
‡
G
m
for
the
[multiplicative]
k-group
schemes
given
by
copies
of
G
m
,
equipped
with
the
respective
labels
“†”,
“‡”.
Thus,
†
q
−1
∈
†
G
m
(k)
(⊆
†
A
1
(k)),
‡
q
∈
‡
G
m
(k)
(⊆
‡
1
A
(k)).
(ii)
Relative
to
the
notation
of
(i),
we
observe
that
∼
(ii-a)
there
exists
a
unique
isomorphism
of
k-ring
schemes
†
A
1
→
‡
A
1
,
but
that
(ii-b)
the
pairs
(
†
A
1
,
†
q
−1
)
and
(
‡
A
1
,
‡
q)
are
not
isomorphic,
i.e.,
as
pairs
consisting
of
a
k-ring
scheme
equipped
with
a
k-rational
point
[cf.
our
assumption
that
q
3
=
q].
By
contrast,
∼
(ii-c)
there
exists
a
unique
isomorphism
of
pairs
(
†
G
m
,
†
q
−1
)
→
(
‡
G
m
,
‡
q),
i.e.,
of
pairs
consisting
of
a
[multiplicative]
k-group
scheme
equipped
with
a
k-rational
point.
∼
Here,
we
observe
that
the
isomorphism
(
†
G
m
,
†
q
−1
)
→
(
‡
G
m
,
‡
q)
of
(ii-c)
does
not
∼
extend
[cf.
(ii-b)!]
to
an
isomorphism
(
†
A
1
,
†
q
−1
)
→
(
‡
A
1
,
‡
q).
In
particular,
∼
(ii-d)
the
isomorphism
of
[multiplicative]
k-group
schemes
†
G
m
→
‡
G
m
is
not
compatible
with
the
k-ring
scheme
structures
of
†
A
1
(⊇
†
G
m
),
‡
1
A
(⊇
‡
G
m
).
Next,
we
observe
that
(ii-e)
the
standard
construction
of
the
projective
line
may
be
regarded
as
the
result
of
gluing
(
†
A
1
,
†
q
−1
)
to
(
‡
A
1
,
‡
q)
along
the
isomorphism
∼
(
†
G
m
,
†
q
−1
)
→
(
‡
G
m
,
‡
q)
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
59
∼
of
(ii-c);
thus,
relative
to
this
gluing,
†
G
m
→
‡
G
m
may
be
regarded
si-
multaneously
as
an
open
subscheme
of
†
A
1
AND
[cf.
“∧”!]
as
an
open
subscheme
of
‡
A
1
.
In
particular,
(ii-d),
(ii-e)
may
be
summarized
as
follows:
the
standard
construction
of
the
projective
line
may
be
regarded
as
a
gluing
of
two
ring
schemes
along
an
isomorphism
of
multiplicative
group
schemes
that
is
not
compatible
with
the
ring
scheme
struc-
tures
on
either
side
of
the
gluing.
Moreover,
we
note
that,
relative
to
this
gluing,
(ii-f)
the
notion
of
a
regular
function
on
†
A
1
cannot
be
expressed
di-
rectly
in
terms
of
the
notion
of
a
regular
function
on
‡
A
1
,
whereas
(ii-g)
the
notion
of
a
rational
function
on
†
A
1
can
be
expressed
directly
in
terms
of
—
i.e.,
in
essence,
coincides,
relative
to
the
above
gluing,
with
—
the
notion
of
a
rational
function
on
‡
A
1
.
Finally,
we
observe
that
(ii-h)
if,
in
the
gluing
of
(ii-e),
one
arbitrarily
deletes
the
distinct
labels
“†”,
“‡”,
then
the
resulting
“gluing
without
labels”
amounts
to
a
gluing
of
a
single
copy
of
A
1
to
itself
that
maps
the
standard
coordinate
“T
”
on
A
1
[regarded,
say,
as
a
rational
function
on
A
1
]
to
T
−1
;
that
is
to
say,
such
a
“gluing
without
labels”
results
in
a
contradiction
[i.e.,
since
T
=
T
−1
!],
unless
one
passes
to
some
sort
of
quotient
of
A
1
—
which
amounts,
from
a
foundational/logical
point
of
view,
to
the
introduction
of
some
sort
of
indeterminacy,
i.e.,
to
the
consideration
of
some
sort
of
collection
of
possibilities
[cf.
“∨”!].
(iii)
The
discussion
of
the
projective
line
in
(ii)
is
truly
remarkable
in
that
it
completely
parallels
—
i.e.,
relative
to
the
correspondence
“−1”
←→
“j
2
”
between
the
exponent
“−1”
in
the
discussion
of
(ii)
and
the
exponents
“j
2
”,
where
j
ranges
from
1
to
l
,
in
the
discussion
of
§3.4
—
numerous
aspects
of
the
Θ-link
of
inter-universal
Teichmüller
theory,
which
we
shall
discuss
in
more
detail
in
§3
[cf.,
especially,
§3.4].
Indeed,
(iii-a)
the
isomorphism
of
(ii-a)
may
be
understood
as
corresponding
to
the
fact
that
the
(Θ
±ell
NF-)Hodge
theaters
on
either
side
of
the
Θ-link
in
inter-universal
Teichmüller
theory
are
isomorphic,
while
(iii-b)
the
observation
of
(ii-b)
may
be
understood
as
corresponding
to
the
fact
that
there
is
no
isomorphism
of
(Θ
±ell
NF-)Hodge
theaters
as
in
(iii-
a)
that
maps
the
Θ-pilot
in
the
domain
of
the
Θ-link
[which
corresponds
to
“
†
q
−1
”]
to
the
q-pilot
in
the
codomain
of
the
Θ-link
[which
corresponds
to
“
‡
q”].
On
the
other
hand,
60
SHINICHI
MOCHIZUKI
(iii-c)
the
isomorphism
of
(ii-c)
may
be
understood
as
corresponding
to
the
full
poly-isomorphism
of
[multiplicative!]
F
×μ
-prime-strips
that
constitutes
the
Θ-link,
while
(iii-d)
the
observation
of
(ii-d)
may
be
understood
as
corresponding
to
the
fact
that
the
full
poly-isomorphism
of
(iii-c)
is
not
compatible
with
the
ring
structures
determined
by
the
(Θ
±ell
NF-)Hodge
theaters
on
either
side
of
the
Θ-link,
i.e.,
in
particular,
does
not
arise
from
a
poly-isomorphism
between
these
(Θ
±ell
NF-)Hodge
theaters
on
either
side
of
the
Θ-link
[cf.
(iii-b)].
Next,
we
observe
that
(iii-e)
the
gluing
of
(ii-e)
may
be
understood
as
corresponding
to
the
glu-
ing
constituted
by
the
Θ-link
between
the
(Θ
±ell
NF-)Hodge
theaters
on
either
side
of
the
Θ-link,
i.e.,
a
gluing
along
[multiplicative!]
F
×μ
-
prime-strips
that
is
not
compatible
with
the
ring
structures
in
the
domain
and
codomain
of
the
Θ-link,
but
which
allows
one
to
obtain
a
single
F
×μ
-prime-strip,
up
to
isomorphism,
that
may
be
interpreted
simultaneously
as
the
F
×μ
-prime-strip
arising
from
the
Θ-pilot
in
the
domain
of
the
Θ-link
AND
[cf.
“∧”!]
as
the
F
×μ
-prime-strip
arising
from
the
q-pilot
in
the
codomain
of
the
Θ-link.
Here,
we
recall
that
this
crucial
logical
AND
“∧”
property
of
the
Θ-link
is
the
central
theme
of
the
present
paper
[cf.
the
discussion
of
Examples
2.4.1,
2.4.2,
2.4.3,
2.4.4,
2.4.5,
2.4.6!].
Next,
we
observe
that
(iii-f)
the
observation
of
(ii-f)
may
be
understood
as
corresponding
to
the
fact
that,
at
least
from
an
a
priori
point
of
view,
there
is
no
natural
way
to
express
the
Θ-pilot
of
the
(Θ
±ell
NF-)Hodge
theater
in
the
domain
of
the
Θ-link,
relative
to
the
gluing
of
(iii-e),
in
terms
of
the
(Θ
±ell
NF-)Hodge
theater
in
the
codomain
of
the
Θ-link,
while
(iii-g)
the
observation
of
(ii-g)
may
be
understood
as
corresponding
to
the
si-
multaneous
holomorphic
expressibility
(SHE)
property
of
the
mul-
tiradial
representation
of
the
Θ-pilot
[cf.
[IUTchIII],
Remark
3.11.1,
(iii);
[IUTchIII],
Remark
3.9.5,
(viii),
(ix)],
which
allows
one
to
express
the
Θ-pilot
of
the
(Θ
±ell
NF-)Hodge
theater
in
the
domain
of
the
Θ-link,
relative
to
the
gluing
of
(iii-e),
in
terms
of
the
(Θ
±ell
NF-)Hodge
theater
in
the
codomain
of
the
Θ-link
[cf.
also
the
discussion
of
(iv),
(v),
below;
the
discussion
surrounding
Example
2.4.8,
(iii-a),
(iii-b),
below].
Finally,
we
note
that
(iii-h)
the
“gluing
without
labels”
discussed
in
(ii-h)
may
be
understood
as
cor-
responding
to
the
oversimplified
version
“RCS-IUT”
of
inter-universal
Teichmüller
theory
obtained
by
implementing
the
RCS-identifications
of
RCS-redundant
copies
of
objects
[cf.
the
discussion
of
§1.2,
Example
2.4.5],
which
leads
to
an
immediate
contradiction,
unless
one
introduces
some
sort
of
quotient/indeterminacy,
i.e.,
which
amounts
to
the
con-
sideration
of
some
sort
of
collection
of
possibilities
[cf.
“∨”!].
In
particular,
relative
to
this
remarkably
close
structural
resemblance
between
the
gluing
that
appears
in
the
standard
construction
of
the
projective
line
and
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
61
the
gluing
constituted
by
the
Θ-link
of
inter-universal
Teichmüller
theory,
the
central
assertion
“IUT
⇔
RCS-IUT”
of
the
RCS
[cf.
the
discussion
of
Example
2.4.5]
may
be
understood
as
correspond-
ing
to
the
assertion
of
an
“obvious
equivalence”
[cf.
the
discussion
of
§1.3]
between
·
the
projective
line,
on
the
one
hand,
and
·
the
affine
line
regarded
up
to
some
sort
of
identification
of
the
standard
coordinate
“T
”
on
the
affine
line
with
its
inverse,
on
the
other.
(iv)
From
the
point
of
view
of
the
analogy
discussed
in
(iii)
between
the
gluing
construction
of
the
projective
line
and
inter-universal
Teichmüller
theory,
perhaps
the
closest
nontrivial
analogue,
in
the
case
of
the
projective
line,
to
the
multiradial
representation
of
the
Θ-pilot
in
inter-universal
Teichmüller
theory
is
the
group
of
projective
general
linear
[i.e.,
“P
GL
2
”]
symmetries
of
the
projective
line
[cf.
also
the
discussion
of
(v)
below;
the
discussion
of
Example
3.10.1,
(i),
below].
That
is
to
say,
although
there
is
also
an
analogy,
discussed
in
(iii-g),
with
the
observation
of
(ii-g),
the
content
of
(ii-g)
is
rather
formal/trivial.
By
contrast,
the
P
GL
2
-symmetries
of
the
projective
line
are
somewhat
less
trivial,
especially
from
the
point
of
view
of
the
gluing
construction
of
the
projective
line
discussed
in
(ii)
[cf.
also
the
discussion
of
(v)
below;
the
discussion
of
Example
3.10.1,
(i),
below].
(v)
The
P
GL
2
-symmetries
of
the
projective
line
are,
in
some
sense,
especially
interesting
in
the
case
where
one
takes
k
to
be
the
field
C
of
complex
numbers,
and
one
restricts
to
the
subgroup
P
U
2
⊆
P
GL
2
(C)
given
by
the
image
of
the
unitary
matrices,
i.e.,
the
projective
unitary
group.
Thus,
as
is
well-known,
one
may
think
of
P
U
2
as
the
group
of
isometric
symmetries
of
the
Riemann
surface
associated
to
the
projective
line
over
C
equipped
with
the
Fubini-
Study
metric.
The
underlying
topological
space
of
this
Riemann
surface
may
be
naturally
identified
with
the
sphere
S
2
.
The
geodesics
associated
to
the
Fubini-Study
metric
then
correspond
to
great
circles
on
the
sphere
S
2
[cf.,
e.g.,
the
illustration
of
[GeoSph]].
In
particular,
the
geodesics
that
pass
through
the
north/south
poles
of
S
2
may
be
thought
of
as
lines
of
longitude.
In
the
current
metrized
situation,
it
is
natural
to
think
of
S
2
as
being
obtained
not
via
a
gluing
of
the
complement
of
the
north
pole
to
the
complement
of
the
south
pole
[i.e.,
as
in
(ii-e)],
but
rather
as
being
obtained
via
a
gluing
S
2
⊇
H
+
⊇
E
=
H
+
∩
H
−
def
⊆
H
−
⊆
S
2
of
the
northern
hemisphere
H
+
⊆
S
2
of
S
2
to
the
southern
hemisphere
H
−
⊆
S
2
of
S
2
along
the
equator
E
⊆
S
2
.
Here,
we
note
that
the
gluing
of
(ii-e)
—
which
yields
a
single
rational
function
on
the
projective
line
that
corresponds
simultaneously
to
the
standard
coordinate
“
†
T
”
on
†
A
1
AND
to
the
standard
coordinate
“
‡
T
−1
”
on
‡
1
A
—
may
be
thought
of,
in
the
current
metrized
situation,
as
corresponding
to
the
following
[at
first
glance,
self-contradictory!]
phenomenon:
62
SHINICHI
MOCHIZUKI
(OrFlw)
an
oriented
flow
along
the
equator
—
which
may
be
thought
of
phys-
ically
as
a
sort
of
wind
current
—
that
flows
from
east
to
west
appears
simultaneously
to
be
flowing,
from
the
point
of
view
of
the
northern
hemi-
sphere
H
+
⊆
S
2
,
in
the
clockwise
direction
AND,
from
the
point
of
view
of
the
southern
hemisphere
H
−
⊆
S
2
,
in
the
counterclockwise
direction.
Next,
let
us
recall
that
—
unlike
S
2
!
—
both
H
+
and
H
−
may
be
thought
of
as
closed
discs
in
the
plane.
Thus,
in
summary,
(GdsFlw)
the
geodesic
geometry
of
the
Fubini-Study
metric
—
i.e.,
in
essence,
the
(P
GL
2
(C)
⊇)
P
U
2
-symmetries
of
S
2
—
allow
one,
by
considering
the
geodesic
flow
along
lines
of
longitude,
to
represent,
up
to
a
relatively
mild
distortion,
the
entirety
of
S
2
,
i.e.,
including
H
−
⊆
S
2
,
as
a
sort
of
extension/deformation
of
the
closed
disc
H
+
.
Indeed,
(GdsFlw)
is
precisely
the
principle
that
is
applied
to
represent,
using
lines
of
longitude,
the
globe
[i.e.,
in
the
sense
of
the
surface
of
the
planet
earth]
via
a
rectangular,
planar,
cartesian
map
[i.e.,
in
the
sense
of
cartography]!
Note,
moreover,
that
(NoLbDlt)
although
the
approach
of
(GdsFlw)
gives
rise
to
a
certain
relatively
mild
degree
of
distortion
in
the
representation
of
H
−
in
terms
of
H
+
,
it
does
not
involve
any
sort
of
naive
identification
of
the
closed
discs
H
+
,
H
−
,
i.e.,
any
sort
of
arbitrary
label
deletion,
in
the
style
of
(ii-h).
The
interpretation
discussed
in
(GdsFlw)
and
(NoLbDlt)
of
the
(P
GL
2
(C)
⊇)
P
U
2
-
symmetries
of
S
2
may
be
understood
as
strongly
suggesting
a
nontrivial
analogy
between
these
symmetries
of
S
2
and
the
multiradial
representation
of
the
Θ-
pilot
in
inter-universal
Teichmüller
theory
[cf.
the
analogy
between
multiradiality
and
connections/parallel
transport/crystals
discussed
in
[Alien],
§3.1,
(iv),
(v),
as
well
as
§3.5,
§3.10,
below].
Example
2.4.8:
Gluings
of
rings
along
multiplicative
monoids.
(i)
Let
R
be
an
integral
domain
equipped
with
the
action
of
a
group
G
and
N
a
positive
integer
≥
2.
For
simplicity,
we
assume
that
N
=
1
+
·
·
·
+
1
[i.e.,
the
sum
of
N
copies
of
“1
∈
R”]
determines
a
nonzero
element
of
R.
Write
·
R
⊆
R
for
the
multiplicative
monoid
of
nonzero
elements
of
R;
·
R
R
μ
for
the
quotient
multiplicative
monoid
of
R
by
the
group
of
roots
of
unity
of
R;
·
(R
)
N
⊆
R
,
(R
μ
)
N
⊆
R
μ
for
the
multiplicative
submonoids
consist-
ing
of
the
N
-th
powers
of
elements
of
“(−)”.
Thus,
G
acts
naturally
and
in
a
compatible
fashion
not
only
on
the
ring
R,
but
also
on
the
multiplicative
monoids
R
,
R
μ
,
(R
)
N
,
(R
μ
)
N
.
Also,
we
observe
that
the
N
-th
power
map
on
R
μ
determines
an
isomorphism
of
multiplicative
monoids
equipped
with
actions
by
G
R
μ
∼
→
(R
μ
)
N
(⊆
R
μ
)
that
does
not
arise
from
a
ring
homomorphism,
i.e.,
as
may
be
seen
from
the
fact
that
this
isomorphism
of
multiplicative
monoids
is
not
compatible
with
the
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
63
operation
of
addition
[cf.
our
assumption
that
N
determines
a
nonzero
element
of
R!]
.
(ii)
Let
†
R,
‡
R
be
two
distinct
copies
of
the
integral
domain
R
of
(i),
equipped
with
respective
actions
by
two
distinct
copies
†
G,
‡
G
of
the
group
G
of
(i).
We
shall
use
similar
notation
for
objects
labeled
with
“†”
or
“‡”
to
the
notation
introduced
in
(i)
for
objects
not
equipped
with
such
labels.
Then
(ii-a)
one
may
use
the
isomorphism
of
multiplicative
monoids
arising
from
the
N
-th
power
map
discussed
in
(i)
to
glue
together
†
G
†
R
⊇
(
†
R
)
N
(
†
R
μ
)
N
∼
←
‡
R
μ
‡
R
⊆
‡
R
‡
G
∼
the
ring
†
R
to
the
ring
‡
R
along
the
multiplicative
monoid
(
†
R
μ
)
N
←
‡
R
μ
[cf.
the
discussion
of
Example
2.4.7,
(ii),
(iii)!].
Here,
we
observe
that
this
gluing
is
compatible
with
the
respective
actions
of
†
G,
∼
‡
G
relative
to
the
isomorphism
†
G
→
‡
G
given
by
forgetting
the
labels
“†”,
“‡”,
but
in
this
context,
it
is
of
the
utmost
importance
to
remember
that
(ii-b)
since,
as
observed
in
(i),
the
N
-th
power
map
is
not
compatible
with
∼
the
operation
of
addition
(!),
this
isomorphism
†
G
→
‡
G
may
be
regarded
either
as
an
isomorphism
of
abstract
groups
or
as
an
isomorphism
of
groups
equipped
with
actions
on
certain
multiplicative
monoids,
but
not
as
an
isomorphism
of
groups
equipped
with
actions
on
rings
[i.e.,
†
R,
‡
R],
e.g.,
as
is
the
case
where
†
G,
‡
G
are
taken
to
be
“Galois
groups”
[that
is
to
say,
groups
equipped
with
faithful
actions
on
some
field,
such
as
the
quotient
field
of
†
R
or
‡
R].
In
the
context
of
(ii-b),
we
observe
that,
of
course,
one
may
also
consider
taking
the
point
of
view
that
†
G,
‡
G
are
groups
equipped
with
actions
on
the
diagram
†
R
⊇
(
†
R
)
N
(
†
R
μ
)
N
∼
←
‡
R
μ
‡
R
⊆
‡
R
[consisting
of
various
rings,
multiplicative
monoids,
etc.]
of
(ii-a),
i.e.,
not
just
on
some
isolated
portion
of
the
diagram
such
as
†
R,
(
†
R
μ
)
N
,
‡
R
μ
,
or
‡
R.
(ii-c)
The
fundamental
—
and
indeed
essentially
tautological!
—
problem,
however,
with
this
approach
of
thinking
of
†
G,
‡
G
as
groups
of
automor-
phisms
of
the
diagram
of
the
above
display
is
that
this
approach
yields
a
situation
in
which
one
can
no
longer
consider
[i.e.,
in
the
sense
that
it
is
no
longer
a
well-defined
proposition
to
consider!]
various
isolated
portions
of
the
diagram
[i.e.,
such
as
†
R,
(
†
R
)
N
,
‡
R
μ
,
or
‡
R]
equipped
with
actions
by
†
G,
‡
G
independently
of
the
entire
diagram.
On
the
other
hand,
as
we
shall
see
in
(iii)
below,
the
main
issue
of
interest
sur-
rounding
the
gluing
of
(ii-a)
involves
consideration
of
the
extent
to
which
one
can
start
precisely
from
such
an
isolated
portion
of
the
diagram
—
namely,
the
glued
data
∼
†
G
(
†
R
μ
)
N
←
‡
R
μ
‡
G
—
and
then
proceed
to
reconstruct,
possibly
up
to
relatively
mild
indeterminacies,
some
remaining
portion
of
the
diagram.
Finally,
we
observe
that
the
importance,
in
64
SHINICHI
MOCHIZUKI
the
context
of
inter-universal
Teichmüller
theory,
of
thinking
of
Galois
groups
[not
as
groups
of
automorphisms
of
ring/fields/diagrams
involving
rings
(!),
but
rather]
as
abstract
groups
[i.e.,
as
emphasized
in
the
above
discussion!]
is
reminiscent
of
the
discussion
of
the
issue
of
the
“relative
subordination”
of
group
theory
versus
field
theory
[i.e.,
“group
theory
≫
field
theory”
versus
“field
theory
≫
group
theory”]
in
[Alien],
§4.4,
(i).
(iii)
In
general,
in
the
situation
of
the
gluing
considered
in
(ii-a),
(iii-a)
the
problem
of
describing
the
additive
structure
of
†
R
in
terms
of
the
additive
structure
of
‡
R
—
in
a
fashion
that
is
compatible
with
the
gluing
and
via
a
single
algorithm
that
may
be
applied
to
the
glued
data
to
reconstruct
simultaneously
the
additive
structures
of
both
†
R
and
‡
R
—
seems
to
be
hopelessly
intractable!
The
nontriviality
of
this
problem
may
already
be
seen,
for
instance,
in
the
case
where
one
takes
R
to
be
Z
[i.e.,
the
ring
of
rational
integers].
Indeed,
this
sort
of
problem
may
be
understood
as
(iii-b)
the
starting
point
of
inter-universal
Teichmüller
theory,
where
one
consid-
ers
the
gluing
constituted
by
the
Θ-link
[cf.
the
discussion
of
§3.4
below]
and
the
issue
of
describing
—
in
a
fashion
compatible
with
the
crucial
logical
AND
property
[cf.
the
discussion
of
(iv)
below]
associated
to
this
gluing!
—
certain
portions
of
the
ring/additive
structure
of
the
domain
[i.e.,
labeled
by
“†”]
of
the
Θ-link
in
terms
of
the
ring/additive
structure
of
the
codomain
[i.e.,
labeled
by
“‡”]
of
the
Θ-link
via
a
sin-
gle
algorithm
that
may
be
applied
to
the
glued
data
to
reconstruct
si-
multaneously
the
corresponding
portions
of
the
ring/additive
structure
of
both
the
domain
and
the
codomain
of
the
Θ-link
[cf.
the
discussion
of
the
simultaneous
holomorphic
expressibility
(SHE)
property
in
[IUTchIII],
Remark
3.11.1,
(iii);
[Alien],
§3.7,
(i);
[Alien],
§3.11,
(iv)].
Such
a
description
is
ultimately
achieved
in
inter-universal
Teichmüller
theory
by
means
of
the
multiradial
representation
of
the
Θ-pilot,
which
allows
one
to
reconstruct,
up
to
relatively
mild
indeterminacies,
certain
portions
of
interest
of
the
ring/additive
structure
of
the
domain
of
the
Θ-link
in
terms
of
the
ring/additive
structure
of
the
codomain
of
the
Θ-link
[cf.
the
discussion
of
Example
2.4.7,
(v)]
—
in
a
fashion
that
is
compatible
with
the
gluing
and
via
a
single
algorithm
that
may
be
applied
to
the
glued
data
to
reconstruct
simultaneously
the
corresponding
portions
of
the
ring/additive
structure
of
both
the
domain
and
the
codomain
of
the
Θ-link
—
by
making
use
of
certain
structural
properties
of
the
various
multiplicative
monoids
equipped
with
group
actions
that
appear
in
the
construction
of
the
Θ-link,
as
well
as
certain
highly
nontrivial
anabelian
properties
of
the
underlying
abstract
groups
of
the
various
Galois
groups
that
appear
[cf.
the
discussion
of
(ii-b)
above;
the
discussion
of
§3.2,
§3.8,
below].
In
this
context,
it
is
also
interesting
to
note
that,
when
N
=
p
is
a
prime
number,
the
fact
that
the
Frobenius
morphism
given
by
raising
to
the
power
p
is
a
ring
homomorphism
in
characteristic
p
may
be
interpreted
in
the
following
way:
(iii-c)
even
in
the
situation
of
the
present
discussion
[i.e.,
where
the
ring
R
is
not
of
positive
characteristic!],
the
isomorphism
of
multiplicative
monoids
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
65
obtained
by
raising
to
the
p-th
power
—
i.e.,
the
isomorphism
of
mul-
tiplicative
monoids
that
appears
in
the
gluing
of
(ii-a)
—
may
in
fact
be
regarded
as
being
“simultaneously
compatible”
with
the
additive
structures
in
its
domain
and
codomain
if
one
regards
one’s
computations
as
being
subject
to
the
“indeterminacy”
given
by
working
modulo
p.
Finally,
we
observe
that
this
interpretation
is
reminiscent
of
the
important
analogies
between
inter-universal
Teichmüller
theory,
on
the
one
hand,
and
Frobenius
liftings
and
p-adic
Teichmüller
theory,
on
the
other,
as
discussed
in
[Alien],
§2.4,
§2.5;
[Alien],
§3.3,
(ii)
[cf.
also
the
discussion
of
crystals
in
[Alien],
§3.1,
(v),
as
well
as
§3.5,
§3.10
below].
(iv)
In
the
context
of
the
gluing
of
(ii-a),
we
observe
that
∼
(iv-a)
the
glued
multiplicative
monoid
(
†
R
μ
)
N
←
‡
R
μ
,
regarded
up
to
iso-
morphism,
is
simultaneously
·
the
multiplicative
monoid
(
†
R
μ
)
N
associated
to
the
ring
†
R
AND
·
the
multiplicative
monoid
‡
R
μ
associated
to
the
ring
‡
R.
∼
In
a
similar
vein,
if
one
thinks
of
the
glued
group
†
G
→
‡
G,
regarded
up
to
iso-
morphism,
either
as
an
abstract
group
or
as
a
group
equipped
with
an
action
on
the
glued
multiplicative
monoid,
then
∼
(iv-b)
this
glued
group
†
G
→
‡
G
is
simultaneously
·
the
group
†
G
equipped
with
an
action
on
the
multiplicative
monoid
(
†
R
μ
)
N
associated
to
the
ring
†
R
AND
·
the
group
‡
G
equipped
with
an
action
on
the
multiplicative
monoid
(
†
R
μ
)
N
associated
to
the
ring
‡
R.
The
gluing
given
by
the
Θ-link
involves
entirely
analogous
logical
AND
prop-
erties,
which
are
fundamental
to
the
essential
logical
structure
of
inter-universal
Teichmüller
theory,
as
exposed
in
the
present
paper.
Of
course,
(iv-c)
it
is
always
possible
to
consider
the
situation
in
which
one
deletes
the
labels
“†”,
“‡”,
but
only
at
the
expense
of
sacrificing
these
crucial
log-
ical
AND
properties
(iv-a),
(iv-b),
i.e.,
at
the
expense
of
agreeing
to
work
under
the
assumption
that
the
“glued
data”
is
the
data
associated
to
“†”
OR
the
data
associated
to
“‡”,
but
not
necessarily
both
simultaneously.
Moreover,
once
one
deletes
the
labels
“†”,
“‡”
—
i.e.,
so
that
the
two
copies
of
“R”
are
identified
with
one
another
via
an
isomorphism
of
rings!
—
the
problem
of
describing
the
ring/additive
structure
of
one
copy
in
terms
of
the
ring/additive
structure
of
the
other
copy
[cf.
the
discussion
of
(iii)!]
becomes
“trivial”,
but
this
triviality
is
of
little
interest
since
it
is
achieved
only
at
the
cost
of
sacrificing
the
crucial
logical
AND
properties
in
favor
of
the
[entirely
uninteresting!]
logical
66
SHINICHI
MOCHIZUKI
OR
property
just
described
—
cf.
the
discussion
surrounding
(NeuORInd)
in
§3.4
below,
as
well
as
the
discussion
of
IUT
=
AND-IUT
versus
RCS-IUT/OR-IUT/EssOR-IUT
in
Examples
2.4.5,
(ii),
(iii),
(v),
(vi),
(vii);
2.4.7,
(ii),
(iii),
(v).
(v)
Finally,
we
note
that
the
relationship
between
the
discussion
of
the
present
Example
2.4.8
and
the
numerical
situation
discussed
in
Example
2.4.5,
(ii),
(iii)
[cf.
also
the
discussion
of
the
final
page
and
a
half
of
the
files
“[SS2018-05]”,
“[SS2018-
08]”
available
at
the
website
[Dsc2018]]
may
be
seen
by
considering
the
case
where
R
is
taken
to
be
the
ring
of
rational
integers
Z
[or,
in
fact,
slightly
more
generally,
the
ring
of
integers
“O
F
”
of
a
finite
field
extension
F
of
the
field
of
rational
numbers
Q
—
a
situation
that
may
be
related
to
the
case
of
Z
by
applying
the
multiplicative
norm
map
N
F/Q
:
F
→
Q].
Indeed,
in
the
case
where
R
is
taken
to
be
Z,
one
may
consider
the
“height”
log(|x|)
∈
R
[where
“log”
denotes
the
natural
logarithm
of
a
positive
real
number,
and
“|
−
|”
denotes
the
absolute
value
of
an
element
of
Z]
associated
to
a
nonzero
element
0
=
x
∈
Z.
Then
the
N
-th
power
map
of
(i),
(ii)
corresponds,
after
passing
to
heights,
to
multiplying
real
numbers
by
N
,
i.e.,
in
essence
to
the
situation
considered
in
Example
2.4.5,
(ii),
(iii)
[which
corresponds
to
the
case
where
the
“N
”
of
the
present
discussion
is
taken
to
be
2].
Section
3:
The
logical
structure
of
inter-universal
Teichmüller
theory
In
the
present
§3,
we
give
a
detailed
exposition
of
the
essential
logical
struc-
ture
of
inter-universal
Teichmüller
theory,
with
a
special
focus
on
issues
related
to
RCS-redundancy.
From
a
strictly
rigorous
point
of
view,
this
exposition
assumes
a
substantial
level
of
knowledge
and
understanding
of
the
technical
details
of
inter-
universal
Teichmüller
theory
[which
are
surveyed,
for
instance,
in
[Alien]].
On
the
other
hand,
in
a
certain
qualitative
sense,
the
discussion
of
the
present
§3
may
in
fact
be
understood,
at
a
relatively
elementary
level,
via
the
analogies
that
we
dis-
cuss
with
the
topics
covered
in
§2.
Indeed,
in
this
context,
it
should
be
emphasized
that,
despite
the
relatively
novel
nature
of
the
set-up
of
inter-universal
Teichmüller
theory,
the
essential
mathematical
content
that
lies
at
the
heart
of
all
of
the
issues
covered
in
the
present
§3
concerns
entirely
well-known
mathematics
at
the
advanced
undergraduate
or
beginning
graduate
level
[i.e.,
the
topics
covered
in
§2].
§3.1.
One-dimensionality
via
identification
of
RCS-redundant
copies
Inter-universal
Teichmüller
theory
concerns
the
explicit
description
of
the
re-
lationship
between
various
possible
intertwinings
—
namely,
the
“Θ”-
and
“q-”
intertwinings
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
67
—
between
the
two
underlying
combinatorial/arithmetic
dimensions
of
a
ring
[cf.,
e.g.,
[Alien],
§2.11;
[Alien],
§3.11,
(v),
as
well
as
the
discussion
of
§3.9
below].
There
are
many
different
ways
of
thinking
about
these
two
underlying
combinatorial/arithmetic
dimensions
of
a
ring;
one
way
to
understand
these
two
dimensions
is
to
think
of
them
as
corresponding,
respectively,
to
the
unit
group
and
value
group
of
the
various
local
fields
that
appear
as
completions
of
a
number
field
at
one
of
its
valuations.
In
more
technical
language,
this
sort
of
decomposition
into
unit
groups
and
value
groups
may
be
seen
in
the
F
×μ
-prime-strips
that
appear
in
the
Θ-link
of
inter-universal
Teichmüller
theory.
Thus,
if
one
thinks
in
terms
of
such
F
×μ
-
prime-strips,
then
inter-universal
Teichmüller
theory
may
be
summarized
as
follows:
(2-Dim)
The
main
content
of
inter-universal
Teichmüller
theory
consists
of
an
explicit
description,
up
to
certain
relatively
mild
indeterminacies,
of
the
Θ-intertwining
on
the
[two-dimensional!]
F
×μ
-prime-strips
that
appear
in
the
Θ-link
in
terms
of
the
q-intertwining
on
these
F
×μ
-
prime-strips
by
means
of
the
log-link
and
various
types
of
Kummer
theory
that
are
used
to
relate
Frobenius-like
and
étale-like
structures
[cf.
the
discussion
of
Example
2.4.8,
(iii)].
In
particular,
the
essential
mathematical
content
of
inter-universal
Teichmüller
the-
ory
concerns
an
a
priori
variable
relationship
between
the
two
underlying
com-
binatorial/arithmetic
dimensions
of
a
ring.
Put
another
way,
if
one
arbitrarily
“crushes”
these
two
dimensions
into
a
single
dimension
—
i.e.,
in
more
technical
language,
assumes
that
(1-Dim)
there
exists
a
consistent
choice
of
a
fixed
relationship
between
these
two
dimensions
of
(2-Dim),
so
that
these
two
dimensions
may,
in
effect,
be
regarded
as
a
single
dimension
—
then
one
immediately
obtains
a
superficial
contradiction
[cf.
the
discussion
of
Example
3.1.1,
(i-b),
(ii-b),
below].
Indeed,
this
is
one
of
the
central
assertions
of
the
RCS
[cf.
the
discussion
following
Example
3.1.1].
This
is
not
a
“new”
observation,
but
rather,
in
some
sense,
the
starting
point
of
inter-universal
Teichmüller
theory,
i.e.,
the
initial
motivation
for
regarding
the
relationship
between
the
two
underlying
combinatorial/arithmetic
dimensions
of
a
ring
as
being
variable,
rather
than
fixed.
Example
3.1.1:
Elementary
models
of
gluings
and
intertwinings.
In
the
following,
we
shall
write
V
for
the
topological
group
R
>0
.
Let
x,
y
∈
V
be
[not
necessarily
distinct!]
elements
of
V
and
Y
⊆
V
a
nonempty
subset
of
V
.
Let
V
rl
,
V
im
be
two
distinct
labeled
copies
of
V
,
which
we
think
of
as
corresponding
to
the
positive
portions
of
the
real
and
imaginary
axes
in
the
complex
plane.
(i)
Let
†
V
,
‡
V
be
two
not
necessarily
distinct
copies
of
V
.
We
shall
write
†
y
∈
†
V
,
‡
x
∈
‡
V
for
the
respective
elements
determined
by
y,
x
∈
V
.
(i-a)
Suppose
that
†
V
,
‡
V
are
distinct
copies
of
V
.
Write
W
for
the
topolog-
ical
space
obtained
by
gluing
†
V
,
‡
V
along
the
homeomorphic
subspaces
{
†
y}
⊆
†
V
,
{
‡
x}
⊆
‡
V
.
Then
observe
that
this
construction
of
W
is
well-
defined
and
free
of
any
internal
contradictions.
Moreover,
the
existence
of
W
does
not
imply
any
nontrivial
conclusions
concerning
x
and
y.
68
SHINICHI
MOCHIZUKI
Note
the
sharp
contrast
between
the
situation
discussed
in
(i-a)
and
the
following
situation:
def
(i-b)
Suppose
that
†
V
,
‡
V
are
in
fact
the
same
copy
of
V
,
i.e.,
∗
V
=
†
V
=
‡
V
.
Consider
the
assertion
that
the
topological
space
∗
V
is
obtained
by
gluing
†
V
,
‡
V
along
the
homeomorphic
subspaces
{
†
y}
⊆
†
V
,
{
‡
x}
⊆
‡
V
.
Then
observe
that
this
assertion
concerning
∗
V
is
well-defined
and
free
of
internal
contradictions
only
in
the
case
where
x
=
y.
That
is
to
say,
the
existence
of
a
topological
space
∗
V
as
described
in
the
above
assertion
implies
the
nontrivial
conclusion
that
x
=
y,
or,
equivalently,
a
“con-
tradiction”
to
the
assertion
that
x
=
y.
One
may
also
consider
the
following
variant
of
(i-b):
(i-c)
One
replaces
{
†
y}
⊆
†
V
in
(i-b)
by
the
nonempty
subset
†
Y
⊆
†
V
[i.e.,
determined
by
Y
⊆
V
],
where
one
thinks
of
this
subset
as
a
set
of
“possible
y’s”.
The
resulting
“assertion”
then
becomes
a
corresponding
collection
of
assertions
related
by
logical
OR
“∨’s”,
and
the
final
nontrivial
con-
clusion
is
that
x
∈
Y
.
(ii)
The
elementary
models
presented
in
(i)
may
be
interpreted
as
essentially
equivalent
representations
of
various
models
of
“holomorphic
structures”
[cf.
the
discussion
below
of
(InfH),
as
well
as
Examples
3.3.1,
3.3.2]
—
i.e.,
in
the
ter-
minology
of
the
discussion
preceding
the
present
Example
3.1.1,
“intertwinings”
—
between
the
“real”
and
“imaginary”
dimensions
V
rl
,
V
im
.
Here,
we
think
of
“holomorphic
structures”/“intertwinings”
as
being
defined
by
assignments
V
rl
1
rl
→
?
∈
V
im
[where
1
rl
∈
V
rl
denotes
the
element
determined
by
1
∈
V
],
corresponding
√
to
“coun-
terclockwise
rotations
by
90
degrees”,
or,
alternatively,
“multiplication
by
−1”.
In-
deed,
let
†
V
rl
,
‡
V
rl
be
two
not
necessarily
distinct
copies
of
V
rl
;
†
V
im
,
‡
V
im
two
not
necessarily
distinct
copies
of
V
im
.
We
shall
write
†
y
im
∈
†
V
im
,
‡
x
im
∈
‡
V
im
for
the
respective
elements
determined
by
y,
x
∈
V
.
Then
the
discussion
of
(i-a)
may
be
translated
into
a
discussion
concerning
intertwinings
by
arguing
as
follows:
(ii-a)
Suppose
that
†
V
rl
,
‡
V
rl
are
distinct
copies
of
V
rl
;
†
V
im
,
‡
V
im
are
distinct
copies
of
V
im
.
Here,
we
think
of
†
V
rl
,
†
V
im
as
being
equipped
with
the
intertwining
given
by
taking
“?”
to
be
†
y
im
∈
†
V
im
;
we
think
of
‡
rl
‡
im
V
,
V
as
being
equipped
with
the
intertwining
given
by
taking
“?”
to
be
‡
x
im
∈
‡
V
im
.
Then
one
applies
(i-a),
relative
to
the
correspondences
†
im
V
←→
†
V
,
‡
V
im
←→
‡
V
.
This
yields
a
gluing
as
in
(i-a)
that
is
well-
defined
and
free
of
any
internal
contradictions.
Moreover,
the
existence
of
such
a
gluing
does
not
imply
any
nontrivial
conclusions
concerning
x
and
y.
In
a
similar
vein:
def
(ii-b)
Suppose
that
†
V
rl
,
‡
V
rl
are
in
fact
the
same
copy
of
V
rl
,
i.e.,
∗
V
rl
=
†
rl
V
=
‡
V
rl
,
and
that
†
V
im
,
‡
V
im
are
in
fact
the
same
copy
of
V
im
,
i.e.,
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
∗
69
def
V
im
=
†
V
im
=
‡
V
im
.
Then
one
applies
(i-b),
relative
to
the
correspon-
dence
∗
V
im
←→
∗
V
.
This
yields
an
assertion
concerning
a
gluing
as
in
(i-b)
—
i.e.,
in
the
language
of
the
present
discussion,
concerning
a
coin-
cidence
of
the
intertwining
on
†
V
rl
,
†
V
im
with
the
intertwining
on
‡
V
rl
,
‡
im
V
—
that
is
well-defined
and
free
of
internal
contradictions
only
in
the
case
where
x
=
y.
That
is
to
say,
the
existence
of
such
a
gluing
implies
the
nontrivial
conclusion
that
x
=
y,
or,
equivalently,
a
“contradiction”
to
the
assertion
that
x
=
y.
(ii-c)
One
replaces
{
†
y
im
}
⊆
†
V
im
in
(ii-b)
[cf.
also
the
notation
of
(ii-a)]
by
the
nonempty
subset
†
Y
im
⊆
†
V
im
[i.e.,
the
subset
determined
by
Y
⊆
V
],
where
one
thinks
of
this
subset
as
a
set
of
“possible
y’s”.
The
resulting
“assertion”
then
becomes
a
corresponding
collection
of
assertions
related
by
logical
OR
“∨’s”,
and
the
final
nontrivial
conclusion
is
that
x
∈
Y
.
(iii)
Relative
to
the
analogy
with
inter-universal
Teichmüller
theory,
we
have
correspondences
with
objects
that
appear
in
the
elementary
models
of
(ii)
as
follows:
V
rl
←→
the
value
group
portion
of
an
F
×μ
-prime-strip;
V
im
←→
the
unit
group
portion
of
an
F
×μ
-prime-strip;
†/‡
←→
(Θ
±ell
NF-)Hodge
theaters
in
the
domain/codomain
of
the
Θ-link;
intertwinings
involving
“y”
←→
the
Θ-intertwinings;
intertwinings
involving
“x”
←→
the
q-intertwinings;
[cf.
the
discussion
at
the
beginning
of
§3.4].
Here,
we
note
that
from
the
point
of
view
of
intertwinings,
the
unit
group
portion
corresponding
to
“V
im
”
must
be
understood
as
being
log-shifted
by
−1,
relative
to
the
value
group
portion
corre-
sponding
to
“V
rl
”
[cf.
the
discussion
below
of
(InfH),
as
well
as
Examples
3.3.1,
3.3.2].
That
is
to
say,
if
the
value
group
portion
corresponding
to
“V
rl
”
is
located
at
the
coordinate
(n,
m)
of
the
log-theta-lattice,
then
the
unit
group
portion
corre-
sponding
to
“V
im
”
must
be
understood
as
being
located
at
the
coordinate
(n,
m−1)
of
the
log-theta-lattice.
In
particular,
the
unit
group
and
value
group
portions
cor-
responding
to
a
pair
“(V
rl
,
V
im
)”
belong
to
different
F
×μ
-prime-strips.
From
the
point
of
view
of
the
discussion
of
(1-Dim),
the
“consistent
choice
of
a
fixed
relationship”
corresponds
to
the
coincidence
of
intertwinings
in
(ii-b),
while
the
resulting
“superficial
contradiction”
corresponds
to
the
“contradiction”
discussed
in
(ii-b).
On
the
other
hand,
the
“explicit
description”/“variable
relationship”
of
(2-Dim),
which
leads
naturally
to
a
numerical
estimate/inequality
concerning
log-
volumes
[cf.
Example
2.4.5,
(ii)],
corresponds
to
the
situation
involving
various
possibilities
discussed
in
(ii-c),
which
leads
to
the
nontrivial
conclusion
“x
∈
Y
”
[cf.
the
discussion
of
“closed
loops”
in
(Stp7),
(Stp8)
of
§3.10
below;
the
discus-
sion
of
(DltLb)
in
§3.11
below;
the
discussion
of
[IUTchIII],
Remark
3.12.2,
(ii)].
(iv)
Finally,
we
observe
in
passing
that
the
fixed
intertwining
of
(ii-b)
[cf.
also
the
discussion
of
(ii-b)
in
(iii),
as
well
as
the
discussion
of
(FxRng),
(FxEuc),
(FxFld),
(RdVar)
below]
may
be
regarded
as
being
analogous
to
the
well-known
classical
holomorphic
approach
to
the
theory
of
moduli
of
[one-dimensional]
70
SHINICHI
MOCHIZUKI
complex
tori,
that
is
to
say,
in
which
one
works
with
a
copy
of
the
upper
half-
plane
“H”
with
a
fixed
holomorphic
structure
and
thinks
of
the
moduli
of
complex
tori
as
a
“variation
of
period
matrices”
[i.e.,
the
holomorphic
parameter
“z
∈
H”,
which
may
be
taken,
in
the
notation
of
(ii-b),
to
be
“ix”
or
“iy”].
By
contrast,
the
situation
involving
the
set
“
†
Y
im
⊆
†
V
im
”
discussed
in
(ii-c)
may
be
regarded
as
analogous
to
the
[real
analytic]
Teichmüller
approach
to
the
theory
of
moduli
of
complex
tori
[cf.
the
discussion
of
Example
3.3.1],
i.e.,
in
which
the
holomorphic
structure
is
subject
to
Teichmüller
dilations
[corresponding
to
various
elements
in
the
set
†
Y
im
],
relative
to
the
fixed
“real
analytic”
pair
given
by
†
V
rl
,
†
V
im
.
One
central
assertion
of
the
RCS
—
which
appears,
for
instance,
in
certain
10pp.
manuscripts
written
by
adherents
of
the
RCS
[cf.,
especially,
the
discussion
of
the
final
page
and
a
half
of
the
files
“[SS2018-05]”,
“[SS2018-08]”
available
at
the
website
[Dsc2018]]
—
is
to
the
effect
that
the
existence,
as
in
(1-Dim),
of
a
consistent
choice
of
a
fixed
relationship
between
the
two
dimensions
of
(2-Dim)
may
be
derived
as
a
consequence
—
i.e.,
in
more
succinct
notation,
(RC-FrÉt),
(RC-log),
(RC-Θ)
“
=⇒
”
(1-Dim)
—
of
certain
“redundant
copies
assertions”,
as
follows:
(RC-FrÉt)
the
Frobenius-like
and
étale-like
versions
of
objects
in
inter-universal
Teichmüller
theory
are
“redundant”,
i.e.,
may
be
identified
with
one
another
without
affecting
the
essential
logical
structure
of
the
theory;
(RC-log)
the
(Θ
±ell
NF-)Hodge
theaters
on
either
side
of
the
log-link
in
inter-
universal
Teichmüller
theory
are
“redundant”,
i.e.,
may
be
identified
with
one
another
without
affecting
the
essential
logical
structure
of
the
theory;
(RC-Θ)
the
(Θ
±ell
NF-)Hodge
theaters
on
either
side
of
the
Θ-link
in
inter-
universal
Teichmüller
theory
are
“redundant”,
i.e.,
may
be
identified
with
one
another
without
affecting
the
essential
logical
structure
of
the
theory.
In
the
remainder
of
the
present
§3
[cf.,
especially,
§3.2,
§3.3,
§3.4],
we
discuss
in
more
detail
the
falsity
of
each
of
these
“RCS-redundancy”
assertions
[i.e.,
(RC-FrÉt),
(RC-log),
(RC-Θ)].
Here,
it
should
be
noted
that
this
falsity
of
(RC-FrÉt),
(RC-log),
(RC-Θ)
is
by
no
means
a
difficult
or
subtle
issue,
but
rather
a
sort
of
matter
of
“belaboring
the
intuitively
obvious”
from
the
point
of
view
of
mathematicians
who
are
thor-
oughly
familiar
with
inter-universal
Teichmüller
theory.
Nevertheless,
as
discussed
in
[Rpt2018],
§17,
it
is
a
pedagogically
meaningful
exercise
to
write
out
and
discuss
the
details
surrounding
this
sort
of
issue.
Moreover,
as
discussed
in
§1.5
of
the
present
paper,
it
is
desirable
from
a
historical
point
of
view
to
produce
detailed,
explicit,
and
readily
accessible
written
expositions
concerning
this
sort
of
issue.
This
state
of
affairs
prompts
the
following
question:
Why
do
adherents
of
the
RCS
continue
to
insist
on
asserting
the
validity
of
these
assertions
(RC-FrÉt),
(RC-log),
(RC-Θ)?
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
71
Any
sort
of
complete,
definitive
answer
to
this
question
lies
beyond
the
scope
of
the
present
paper.
On
the
other
hand,
it
seems
natural
to
conjecture
that
one
fundamental
motivation
for
these
assertions
of
RCS-redundancy
may
be
found
in
the
fact
that
(FxRng)
many
arithmetic
geometers
have
only
experienced
working
in
situations
where
all
schemes
—
or,
alternatively,
rings
—
that
appear
in
a
theory
are
regarded
as
belonging
to
a
single
category
that
is
fixed
throughout
the
theory,
hence
are
related
to
another
via
ring
homomorphisms,
i.e.,
in
such
a
way
that
the
ring
structure
of
the
various
rings
involved
is
always
respected
[cf.
the
discussion
of
§1.5,
as
well
as
the
discussion
of
§3.8
below].
It
is
not
difficult
to
imagine
that
the
heuristics
and
intuition
that
result
from
years
[or
decades!]
of
immersive
experience
in
such
mathematical
situations
could
create
a
mindset
that
is
fertile
ground
for
the
RCS-redundancy
assertions
that
will
be
discussed
in
detail
in
the
remainder
of
the
present
§3
[cf.,
especially,
§3.2,
§3.3,
§3.4].
Finally,
we
observe
that
this
situation
is,
in
certain
respects,
reminiscent
of
various
situations
that
occurred
throughout
the
history
of
mathematics,
such
as,
for
instance,
the
situation
that
occurred
in
the
late
19-th
century
with
regard
to
such
novel
[i.e.,
at
the
time]
notions
as
the
notion
of
an
abstract
manifold
or
an
abstract
Riemann
surface.
That
is
to
say,
(FxEuc)
from
the
point
of
view
of
anyone
for
whom
it
is
a
“matter
of
course”
or
“common
sense”
that
all
geometry
must
take
place
within
some
fixed,
static
ambient
Euclidean
space
—
such
as,
for
instance,
the
complex
plane
—
such
abstract
geometric
notions
as
the
notion
of
an
abstract
manifold
or
abstract
Riemann
surface
might
come
across
as
deeply
dis-
turbing
and
unlikely
to
be
of
use
in
any
substantive
mathematical
sense
[cf.
the
discussion
of
§1.5;
the
discussion
of
[IUTchI],
§I2].
In
this
context,
it
is
of
interest
—
especially
from
a
historical
point
of
view
—
to
recall
that,
in
some
sense,
the
most
fundamental
classical
example
of
such
an
abstract
geometry
is
the
Riemann
surface
that
arises
by
applying
the
technique
of
analytic
continuation
to
the
complex
logarithm,
i.e.,
which
may
be
regarded
as
a
sort
of
distant
ancestor
[cf.
the
discussion
of
[IUTchI],
Remark
5.1.4;
[Alien],
§3.3,
(vi)]
of
the
log-link
of
inter-universal
Teichmüller
theory.
Another
[in
fact
closely
related!]
fundamental
classical
example
of
such
an
abstract
geometry
is
the
hyperbolic
geometry
of
the
upper
half-plane,
which
may
also
be
regarded
as
a
sort
of
distant
ancestor
of
numerous
aspects
of
inter-universal
Teichmüller
theory
[cf.
(InfH)
and
Example
3.3.2
in
§3.3
below,
as
well
as
the
discussion
of
[IUTchI],
Remark
6.12.3,
(iii);
[IUTchIII],
Remark
2.3.3,
(ix),
(x)].
Another
historically
important
instance
of
this
sort
of
situation
may
be
seen
in
the
introduction,
in
the
early
19-th
century,
of
Galois
groups
—
i.e.,
of
[finite]
automorphism
groups
of
abstract
fields
—
as
a
tool
for
investigating
the
roots
of
polynomial
equations.
That
is
to
say,
(FxFld)
until
the
advent
of
Galois
groups/abstract
fields,
the
issue
of
investigat-
ing
the
roots
of
polynomial
equations
was
always
regarded
—
again
as
a
72
SHINICHI
MOCHIZUKI
“matter
of
course”
or
“common
sense”
—
as
an
issue
of
investigating
var-
ious
“exotic
numbers”
inside
some
fixed,
static
ambient
field
such
as
the
field
of
complex
numbers;
moreover,
from
this
more
classical
“common
sense”
point
of
view,
the
idea
of
working
with
automorphisms
of
abstract
fields
—
i.e.,
fields
that
are
not
constrained
[since
such
constraints
would
rule
out
the
existence
of
nontrivial
automorphisms!]
to
be
treated
as
sub-
sets
of
some
fixed,
static
ambient
field
—
might
come
across
as
deeply
disturbing
and
unlikely
to
be
of
use
in
any
substantive
mathematical
sense
[cf.
the
discussion
of
§1.5].
On
the
other
hand,
from
the
point
of
view
of
inter-universal
Teichmüller
theory,
this
radical
transition
roots
as
concrete
numbers
Galois
groups/abstract
fields
that
occurred
in
the
early
19-th
century
may
be
regarded
as
a
sort
of
distant
ancestor
of
the
transition
Galois
groups/abstract
fields
abstract
groups/anabelian
algorithms
that
occurs
in
inter-universal
Teichmüller
theory
[cf.
the
discussion
at
the
beginning
of
§3.2
below;
the
discussion
of
§3.8
below;
the
discussion
of
the
final
portion
of
[Alien],
§4.4,
(i)].
A
somewhat
more
recent
historical
example
of
this
sort
of
situation
may
be
seen
in
the
situation
surrounding
the
introduction
of
[possibly
non-reduced]
schemes
by
Grothendieck
in
the
early
1960’s.
Indeed,
(RdVar)
the
possible
existence
of
nilpotent
elements
in
the
structure
sheaf
of
a
non-reduced
scheme
struck
many
more
classically
oriented
algebraic
ge-
ometers,
who
were
accustomed
to
working
only
with
reduced
varieties
—
whose
geometry
could
be
understood
intuitively
in
terms
of
their
sets
of
closed
points
—
as
being
entirely
meaningless
and
unlikely
to
be
of
use
in
any
substantive
mathematical
sense,
especially
since
it
was
taken
as
a
“matter
of
course”
or
“common
sense”
that
any
mathematically
sub-
stantive
property
of
a
variety
would
most
certainly
necessarily
be
readily
identifiable
at
the
level
of
the
set
of
closed
points
of
the
variety
[cf.
the
discussion
of
§1.5].
This
more
recent
example
of
non-reduced
schemes
is
especially
of
interest
in
the
context
of
inter-universal
Teichmüller
theory
in
light
of
the
strong
structural
re-
semblances
that
exist
between
the
notion
of
multiradiality
in
inter-universal
Te-
ichmüller
theory
and
the
theory
[due
to
Grothendieck!]
of
crystals
[cf.
[Alien],
§3.1,
(v);
the
discussion
of
(CrAND)
in
§3.5
below;
the
discussion
of
§3.10
below].
Indeed,
the
“trivialization”
of
the
theory
of
crystals
that
results
from
replacing
the
[non-reduced!]
nilpotent
thickenings
that
appear
in
the
theory
of
crystals
by
the
associated
reduced
schemes
corresponds
precisely
to
the
situation
discussed
in
(CrRCS)
[cf.
also
(CrOR)]
in
§3.5
below.
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
73
§3.2.
RCS-redundancy
of
Frobenius-like/étale-like
versions
of
objects
We
begin
by
recalling
that
(Θ
±ell
NF-)Hodge
theaters
—
i.e.,
lattice
points
in
the
log-theta-lattice
—
give
rise
to
both
Frobenius-like
and
étale-like
objects.
Whereas
the
datum
of
a
Frobenius-like
object
depends,
a
priori,
on
the
coordinates
“(n,
m)”
of
the
(Θ
±ell
NF-)Hodge
theater
from
which
it
arises,
étale-like
objects
satisfy
various
[horizontal/vertical]
coricity
properties
to
the
effect
that
they
map
isomorphically
to
corresponding
objects
in
a
vertically
[in
the
case
of
vertical
coricity]
or
horizontally
[in
the
case
of
horizontal
coricity]
neighboring
(Θ
±ell
NF-
)Hodge
theater
of
the
log-theta-lattice
[cf.
the
discussion
of
Example
3.2.2,
(i),
(iv),
below;
[Alien],
§2.7,
(i),
(ii),
(iii),
(iv);
[Alien],
§2.8,
2
Fr/ét
;
[Alien],
§3.3,
(ii),
(vi),
(vii);
[Rpt2018],
§15].
Here,
we
recall
that
étale-like
objects
correspond,
for
the
most
part,
to
arithmetic
fundamental
groups
—
such
as,
for
instance,
the
étale
fundamental
group
“π
1
(X)”
of
a
hyperbolic
curve
X
over
a
number
field
or
mixed
characteristic
local
field
—
or,
more
generally,
to
objects
that
may
be
reconstructed
from
such
arithmetic
fundamental
groups,
so
long
as
the
object
is
regarded
as
being
equipped
with
auxil-
iary
data
consisting
of
the
arithmetic
fundamental
group
from
which
it
was
recon-
structed,
together
with
the
reconstruction
algorithm
that
was
applied
to
reconstruct
the
object.
Here,
we
recall
that,
in
this
context,
it
is
of
fundamental
importance
that
these
arithmetic
fundamental
groups
be
treated
simply
as
abstract
topological
groups
[cf.
the
discussion
of
§3.8
below
for
more
details].
Étale-like
objects
also
satisfy
a
crucial
symmetry
property
with
respect
to
permutation
of
adjacent
vertical
lines
of
the
log-theta-lattice
[cf.
Example
3.2.2,
(ii),
(iv),
below;
[Alien],
§3.2;
the
discussion
surrounding
Fig.
3.12
in
[Alien],
§3.6,
(i)].
That
is
to
say,
in
summary,
the
crucial
coricity/symmetry
properties
satisfied
by
étale-like
objects
—
which
are,
in
essence,
a
formal
consequence
of
treating
the
arithmetic
fundamental
groups
that
appear
as
abstract
topological
groups
[cf.
the
discussion
of
§3.8
below
for
more
details]
—
play
a
central
role
in
the
multiradial
algorithms
of
inter-universal
Teichmüller
theory
[i.e.,
[IUTchIII],
Theorem
3.11]
and
are
not
satisfied
by
Frobenius-like
ob-
jects
—
cf.,
the
discussion
of
Example
3.2.2,
(i),
(ii),
(iv),
below;
[Alien],
§2.7,
(ii),
(iii);
[Alien],
§3.1,
(iii);
[Alien],
Example
3.2.2;
[Rpt2018],
§15,
(LbΘ),
(Lblog),
(LbMn),
(EtFr),
(EtΘ),
(Etlog),
(EtMn).
On
the
other
hand,
once
one
implements
the
RCS-identifications
discussed
in
(RC-log),
(RC-Θ),
there
is,
in
effect,
“only
one”
(Θ
±ell
NF-)Hodge
theater
in
the
log-theta-lattice,
so
all
issues
of
determining
relationships
between
corresponding
objects
in
(Θ
±ell
NF-)Hodge
theaters
at
distinct
coordinates
“(n,
m)”
of
the
log-
theta-lattice
appear,
at
first
glance,
to
have
been
“trivially
resolved”.
Put
another
way,
74
SHINICHI
MOCHIZUKI
once
one
implements
the
RCS-identifications
of
(RC-log),
(RC-Θ),
even
Frobenius-like
objects
appear,
at
first
glance,
to
satisfy
all
possible
coric-
ity/symmetry
properties,
i.e.,
at
a
more
symbolic
level,
(RC-log),
(RC-Θ)
“
=⇒
”
(RC-FrÉt).
In
particular,
the
assertions
of
the
RCS
discussed
in
§3.1
and
the
present
§3.2
may
be
summarized,
at
a
symbolic
level,
as
follows:
(RC-log),
(RC-Θ)
“
=⇒
”
(RC-FrÉt),
(RC-log),
(RC-Θ)
“
=⇒
”
(1-Dim).
In
fact,
however,
the
RCS-identifications
of
(RC-log),
(RC-Θ)
do
not
resolve
such
issues
[i.e.,
of
relating
corresponding
objects
in
(Θ
±ell
NF-)Hodge
theaters
at
distinct
coordinates
“(n,
m)”
of
the
log-theta-lattice]
at
all
[cf.
the
discussion
of
symmetries
in
Example
2.3.1,
(iii)!],
but
rather
merely
have
the
effect
of
translating/reformulating
such
issues
of
relating
corresponding
objects
in
(Θ
±ell
NF-)Hodge
theaters
at
distinct
coordinates
“(n,
m)”
of
the
log-theta-
lattice
into
issues
of
tracking
the
effect
on
objects
in
(Θ
±ell
NF-)Hodge
the-
aters
as
one
moves
along
the
paths
constituted
by
various
composites
of
Θ-
and
log-links.
On
the
other
hand,
at
a
purely
formal
level,
the
discussion
given
above
of
the
falsity
of
(RC-FrÉt)
—
i.e.,
as
a
conse-
quence
of
the
crucial
coricity/symmetry
properties
discussed
above
—
is,
in
some
sense,
predicated
on
the
falsity
of
(RC-log),
(RC-Θ).
This
falsity
of
(RC-log),
(RC-Θ)
will
be
discussed
in
detail
in
§3.3,
§3.4,
below.
In
this
context,
it
is
useful
to
observe
that
the
situation
surrounding
the
Θ-
link
and
(RC-Θ),
(RC-FrÉt)
(respectively,
the
log-link
and
(RC-log),
(RC-FrÉt))
is
structurally
reminiscent
of
the
object
J
discussed
in
Examples
2.3.2,
2.4.1,
2.4.2
[cf.
also
the
correspondences
discussed
in
Example
2.4.5,
(ii);
the
discussion
of
[IUTchIII],
Remark
1.2.2,
(vi),
(vii)],
i.e.,
if
one
regards
(StR1)
the
domain
of
the
Θ-
(respectively,
log-)
link
as
corresponding
to
†
I,
(StR2)
the
codomain
of
the
Θ-
(respectively,
log-)
link
as
corresponding
to
‡
I,
(StR3)
the
gluing
data
—
i.e.,
a
certain
F
×μ
-prime-strip
(respectively,
F-
prime-strip)
—
that
arises
from
the
domain
(Θ
±ell
NF-)Hodge
theater
of
the
Θ-
(respectively,
log-)
link
as
corresponding
to
†
β
=
γ
J
,
(StR4)
the
gluing
data
—
i.e.,
a
certain
F
×μ
-prime-strip
(respectively,
F-
prime-strip)
—
that
arises
from
the
codomain
(Θ
±ell
NF-)Hodge
theater
of
the
Θ-
(respectively,
log-)
link
as
corresponding
to
γ
J
=
‡
α,
(StR5)
the
étale-like
objects
that
are
coric
with
respect
to
the
Θ-
(respectively,
log-)
link
as
corresponding
to
the
glued
differential
discussed
in
(AOD3),
and
(StR6)
the
RCS-identification
of
(RC-Θ)
(respectively,
(RC-log))
as
correspond-
ing
to
the
operation
of
passing
to
the
quotient
J
M
=
J/
†
I
∼
‡
I
∼
→
L
=
I/
α
∼
β.
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
75
This
strong
structural
similarity
will
play
an
important
role
in
the
discussion
of
§3.3,
§3.4,
below.
Finally,
we
observe
that
the
portion,
i.e.,
(StR5),
of
this
strong
structural
similarity
involving
the
glued
differential
discussed
in
(AOD3)
is
particularly
of
interest
in
the
context
of
the
discussion
of
[Alien],
§2.
That
is
to
say,
as
discussed
in
the
first
paragraph
of
[Alien],
§2.6,
étale-like
objects
in
inter-universal
Teichmüller
theory
play
an
analogous
role
to
the
role
played
by
tangent
bundles/sheaves
of
differentials
in
the
(InvHt)
special
case
of
the
invariance
of
the
height
under
isogenies
between
abelian
varieties
[due
to
Faltings]
discussed
in
[Alien],
§2.3,
§2.4
[cf.
also
[Rpt2018],
§16,
(DiIsm),
as
well
as
the
discussion
of
Example
3.2.1
below;
the
discussion
of
§3.5
below],
as
well
as
in
the
(FrDff)
discussion
of
differentiation
of
p-adic
liftings
of
the
Frobenius
mor-
phism
given
in
[Alien],
§2.5.
The
efficacy
of
the
technique
of
considering
induced
maps
on
differentials
in
the
various
examples
discussed
in
[Alien],
§2.3,
§2.4,
§2.5,
may
also
be
observed
in
the
discussion
of
the
fundamental
theorem
of
calculus
in
§2.2,
as
well
as
in
the
context
of
(RC-FrÉt)
and
(StR5).
In
light
of
the
central
importance
of
(InvHt),
as
well
as
the
closely
related
coricity/symmetry/commutativity
properties
of
the
log-theta-lattice
[cf.
the
discussion
at
the
beginning
of
the
present
§3.2,
as
well
as
the
discussion
at
the
beginning
§3.3
below],
in
the
essential
logical
structure
of
inter-universal
Teichmüller
theory,
we
pause
to
give
a
brief
review/exposition
of
(InvHt)
and
these
closely
related
coricity/symmetry/commutativity
properties,
in
the
following
Examples
3.2.1,
3.2.2.
Example
3.2.1:
Global
multiplicative
subspaces
and
bounds
on
heights.
(i)
Let
p
be
a
prime
number,
K
a
finite
extension
of
the
field
Q
p
of
p-adic
numbers,
E
an
elliptic
curve
over
K
with
bad
multiplicative
—
i.e.,
in
other
words,
nonsmooth
semi-stable
—
reduction
over
the
ring
of
integers
O
K
of
K.
Write
m
K
⊆
O
K
for
the
maximal
ideal
of
O
K
.
Thus,
E
is
a
Tate
curve
and
hence
[cf.
the
theory
of
[Mumf2]]
may
be
represented,
using
the
theory
of
formal
schemes,
as
a
sort
of
quotient
Z
”
“G
m
/q
E
of
the
multiplicative
group
scheme
G
m
over
K
by
the
subgroup
generated
by
the
[nonzero]
q-parameter
q
E
∈
m
K
of
the
elliptic
curve
E.
Let
l
be
a
prime
number.
Then
the
subscheme
μ
l
of
l-torsion
points
of
G
m
determines,
via
the
above
p-adic
quotient
representation,
a
canonical
exact
sequence
1
−→
μ
l
−→
E[l]
−→
Z/lZ
−→
1
—
where
we
write
E[l]
⊆
E
for
the
subgroup
scheme
of
l-torsion
points
of
E,
and
we
observe
that
the
generator
“q
E
”
of
the
group
of
deck
transformations
of
the
above
quotient
representation
determines
a
canonical
generator
γ
l
∈
(E[l]/μ
l
)(K),
up
to
multiplication
by
±1,
of
(E[l]/μ
l
)(K),
i.e.,
a
canonical
isomorphism,
up
to
multiplication
by
±1,
of
the
quotient
group
scheme
E[l]/μ
l
with
[the
group
scheme
over
K
determined
by]
Z/lZ.
Thus,
in
summary,
76
SHINICHI
MOCHIZUKI
the
subgroup
scheme
E[l]
⊆
E
of
l-torsion
points
of
E
is
equipped
with
a
canonical
multiplicative
subspace
μ
l
(→
E[l]),
as
well
as
with
a
canonical
generator,
up
to
multiplication
by
±1,
of
the
quotient
E[l]/μ
l
.
(ii)
Let
(E,
0
E
)
be
the
pointed
Riemann
surface
determined
by
an
elliptic
curve
def
over
the
field
of
complex
numbers
C.
Write
E[∞]
=
π
1
top
(E)
for
the
[usual
topologi-
cal]
fundamental
group
of
E,
relative
to
the
basepoint
determined
by
the
origin
0
E
of
the
given
elliptic
curve.
Thus,
E[∞]
is
a
free
abelian
group
on
two
generators.
Let
M
⊆
E[∞]
be
a
rank
one
[free
abelian]
subgroup
such
that
E[∞]/M
is
torsion-free.
Then
M
corresponds
to
an
infinite
covering
E
M
→
E
of
E
such
that
any
point
0
E
M
of
E
M
that
lifts
0
E
determines,
up
to
possible
compo-
∼
sition
with
the
inversion
automorphism,
an
isomorphism
E
M
→
C
×
of
complex
Lie
groups,
relative
to
the
unique
complex
Lie
group
structure
on
the
pointed
Riemann
surface
(E
M
,
0
E
M
)
that
lifts
the
unique
complex
Lie
group
structure
on
the
pointed
Riemann
surface
(E,
0
E
).
In
particular,
we
conclude
that
the
choice
of
M
[together
∼
with
a
choice
of
an
isomorphism
E
M
→
C
×
of
the
sort
just
discussed]
determines
a
complex
holomorphic
quotient
representation
“C
×
/q
E
Z
”
of
the
given
elliptic
curve,
together
with
an
exact
sequence
1
−→
M
−→
E[∞]
−→
Z
−→
1
—
where
we
observe
that
the
rank
one
free
abelian
group
E[∞]/M
(
∼
=
Z)
is
equipped
with
a
unique
choice
of
generator
γ
∞
∈
E[∞]/M,
up
to
multiplication
by
±1.
Thus,
the
p-adic
quotient
representation,
as
well
as
the
associated
exact
sequence
and
canonical
generator
[up
to
multiplication
by
±1],
discussed
in
(i)
may
be
regarded
as
p-adic
analogues
of
the
complex
holomorphic
quotient
representation
and
associated
exact
sequence/canonical
generator
[up
to
multiplication
by
±1]
discussed
in
the
present
(ii).
(iii)
Let
F
be
a
number
field
[i.e.,
a
finite
extension
of
the
field
Q
of
rational
numbers]
and
E
an
elliptic
curve
over
F
.
Write
E[l]
⊆
E
for
the
subgroup
scheme
of
l-torsion
points
of
E.
Then,
in
general,
E
does
not
necessarily
admit
a
global
multiplicative
subspace
(GMS)
M
⊆
E[l]
or
a
global
canonical
generator
(GCG)
γ
∈
(E[l]/M
)(F
),
i.e.,
a
subgroup
scheme
M
⊆
E[l]
or
generator
γ
∈
(E[l]/M
)(F
)
that
restricts
to
the
multiplicative
subspace
or
canonical
generator
discussed
in
(i)
at
each
nonarchimedean
prime
of
F
where
E
has
bad
multiplicative
reduction.
On
the
other
hand,
let
us
suppose,
for
the
remainder
of
the
present
(iii),
that
E
does
admit
a
global
multiplicative
subspace
(GMS)
M
⊆
E[l].
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
77
def
Write
E
∗
=
E/M
.
Thus,
we
have
an
isogeny
φ
:
E
→
E
∗
that,
in
light
of
the
discussion
of
(i),
together
with
the
fact
that
M
⊆
E[l]
is
a
GMS,
corresponds
to
the
isogeny
Z
l·Z
”
−→
“G
m
/q
E
”
“G
m
/q
E
given
by
raising
to
the
l-th
power
on
G
m
at
each
nonarchimedean
prime
of
F
where
E
has
bad
multiplicative
reduction.
In
particular,
at
each
nonarchimedean
prime
v
of
F
where
E
has
bad
multiplicative
reduction,
the
q-parameter
q
E
∗
,v
of
E
∗
at
v
is
the
l-th
power
of
the
q-parameter
q
E,v
of
E
at
v,
i.e.,
l
.
q
E
∗
,v
=
q
E,v
In
particular,
if
we
write
log(q
(−)
)
∈
R
for
the
normalized
arithmetic
degree
“deg(−)”
[cf.
the
discussion
preceding
[GenEll],
Definition
1.2]
of
the
arithmetic
divisor
determined
by
the
q-parameters
of
an
elliptic
curve
over
a
number
field
at
the
nonarchimedean
primes
where
the
elliptic
curve,
which
we
denote
“(−)”,
has
bad
multiplicative
reduction,
then
we
obtain
the
relation
log(q
E
∗
)
=
l
·
log(q
E
)
∈
R
between
log(q
E
),
log(q
E
∗
)
∈
R.
(iv)
We
continue
to
consider
the
situation
discussed
in
(iii).
Write
M
ell
⊇
M
ell
for
the
compactified
moduli
stack
of
elliptic
curves
—
or,
equivalently,
the
moduli
stack
of
pointed
stable
curves
of
type
(1,
1)
—
over
Z
and
the
open
substack
obtained
by
forming
the
complement
of
the
divisor
at
infinity
∞
M
ell
⊆
M
ell
.
Write
ω
M
ell
for
the
ample
line
bundle
on
M
ell
determined
by
the
cotangent
space
at
the
origin
of
the
tautological
family
of
one-dimensional
semi-abelian
schemes
over
M
ell
[i.e.,
obtained
by
forming
the
complement
of
the
unique
node
of
the
tautological
pointed
stable
curve
of
type
(1,
1)
over
M
ell
].
Now
recall
the
discriminant
moduli
form
Δ
M
ell
,
which
may
be
thought
of
as
an
isomorphism
of
line
bundles
O
M
ell
∼
⊗12
ω
M
(−∞
M
ell
)
→
ell
⊗12
—
i.e.,
a
section
of
ω
M
over
M
ell
that
has
no
zeroes
or
poles
except
for
a
zero
of
ell
order
1
at
∞
M
ell
.
It
follows
immediately
from
the
existence
of
Δ
M
ell
[cf.,
e.g.,
the
discussion
of
[GenEll],
§3,
for
more
details]
that,
if,
for
the
sake
of
simplicity,
we
ignore
the
contributions
at
the
archimedean
primes,
then
we
obtain
the
relation
ht
(−)
≈
1
6
log(q
(−)
)
—
where
we
write
ht(−)
for
the
normalized
height
associated
to
the
ample
line
⊗2
on
M
ell
of
the
point
determined
by
an
elliptic
curve
“(−)”
defined
bundle
ω
M
ell
78
SHINICHI
MOCHIZUKI
over
a
number
field
and
“≈”
to
signify
a
relationship
of
bounded
discrepancy
[i.e.,
that
the
absolute
value
of
the
difference
between
the
left-
and
right-hand
sides
is
bounded
by
some
positive
real
number
independently
of
“(−)”].
(v)
We
continue
to
consider
the
situation
discussed
in
(iii),
(iv).
Recall
the
isogeny
φ
:
E
→
E
∗
discussed
in
(iii).
Since
this
isogeny
is
of
degree
l,
hence,
in
particular,
étale
over
nonarchimedean
primes
of
F
of
residue
characteristic
=
l,
we
conclude
immediately,
via
a
straightforward
computation
of
the
map
dφ
induced
on
differentials
by
φ
[cf.
the
proof
of
[GenEll],
Lemma
3.5,
for
more
details],
that,
if,
for
the
sake
of
simplicity,
we
ignore
the
contributions
at
the
archimedean
primes,
then
we
obtain
relations
ht
E
−
log(l)
ht
E
∗
ht
E
+
log(l)
[where
we
use
the
notation
“”
to
signify
an
inequality
“≤”
up
to
bounded
discrep-
ancy,
i.e.,
a
relation
to
the
effect
that
the
left-hand
side
is
bounded,
independently
of
the
elliptic
curve
E
and
the
prime
number
l,
by
the
sum
of
the
right-hand
side
and
some
positive
real
number]
and
hence,
by
combining
the
latter
relation
“”
with
the
relations
of
the
final
displays
of
(iii),
(iv),
that
ht
E
1
l
ht
E
+
log(l)
1
log(l)
≤
1.
That
is
to
say,
in
summary,
if,
for
the
sake
of
—
i.e.,
that
ht
E
l−1
simplicity,
we
ignore
the
contributions
at
the
archimedean
primes,
then
the
assumption
[cf.
(iii)]
that
E
admits
a
GMS
implies
a
bound
on
the
height
of
E
[i.e.,
ht
E
]
and
hence,
in
particular,
that,
if
one
only
considers
number
fields
F
of
bounded
degree
over
Q,
then
there
are
only
finitely
possibilities
for
the
isomorphism
class
of
E.
This
is
precisely
the
argument
given
in
[GenEll],
Lemma
3.5,
which
may
be
regarded
as
a
special
case
of
the
argument
given
in
the
original
proof
[due
to
Faltings]
of
the
invariance
of
the
height
[up
to
bounded
discrepancies]
under
isogenies
of
abelian
varieties.
(vi)
The
argument
reviewed
in
(v)
may
be
understood
as
consisting
of
two
key
points,
both
of
which
are
closely
related
to
various
central
aspects
of
inter-universal
Teichmüller
theory.
The
first
key
point
is,
of
course,
(vi-a)
the
assumption
of
the
existence
of
a
GMS
[cf.
(iii)],
which
implies
that
the
passage
E
E
∗
to
the
quotient
of
E
by
the
GMS
corresponds
to
a
relation
l
q
E
→
q
E
between
the
q-parameters
of
E
and
E
∗
at
each
nonarchimedean
prime
of
F
where
E
and
E
∗
have
bad
multiplicative
reduction
—
i.e.,
to
a
relation
reminiscent
of
the
Frobenius
morphism
in
positive
characteristic.
Here,
we
observe
in
passing
that,
in
the
absence
of
this
crucial
assumption
of
the
existence
of
a
GMS,
the
passage
from
E
to
some
arbitrary
quotient
of
E
by
a
l
”
at
some
finite
subgroup
scheme
of
rank
l
would
give
rise
to
relations
“q
E
→
q
E
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
79
1/l
nonarchimedean
primes
of
bad
multiplicative
reduction
and
to
relations
“q
E
→
q
E
”
at
other
nonarchimedean
primes
of
bad
multiplicative
reduction
—
i.e.,
a
situation
in
which
the
argument
reviewed
in
(v)
would
break
down
completely!
In
inter-
universal
Teichmüller
theory,
(vi-b)
the
fundamental
role
played
by
theta
functions
[cf.
the
discussion
of
Examples
3.3.2,
3.8.4
below]
—
i.e.,
in
the
multiradial
reconstruction
2
algorithms
of
the
Θ-pilot
“{q
j
}”
—
means
that
in
addition
to
a
GMS,
v
it
will
be
necessary
to
somehow
“simulate”
[cf.
the
discussion
of
Example
3.8.2,
(i),
below]
the
existence
of
a
GCG.
Here,
it
is
useful
to
recall
that,
from
the
point
of
view
of
the
classical
complex
theory
of
theta
functions
[cf.
also
Example
3.3.2
below]
+∞
2
q
n
·
U
n
n=−∞
—
where
q,
U
∈
C
and
|q|
<
1:
·
the
“U
”
may
be
understood
as
the
standard
multiplicative
coor-
∼
dinate
on
the
infinite
covering
“C
×
→
E
M
→
E”
of
(ii),
hence
is
only
defined
once
one
has
a
“multiplicative
subspace
M
⊆
E[∞]”,
i.e.,
the
com-
plex
analogue
of
the
l-torsion
multiplicative
subspace
“μ
l
⊆
E[l]”
of
(i),
or,
alternatively,
of
the
coverings
“X
K
”,
“C
K
”
of
[Alien],
§3.3,
(i),
(iv),
(v)
[cf.
also
the
discussion
of
Example
3.8.2,
(i),
below];
·
the
“q”
may
be
understood
as
the
complex
q-parameter
determined
by
∼
a
generating
deck
transformation
of
the
finite
covering
“C
×
→
E
M
→
E”
of
(ii),
hence
is
only
defined
once
one
has
a
“generator,
up
to
multiplication
by
±1,
of
E[∞]/M”,
i.e.,
the
complex
analogue
of
the
l-torsion
canonical
generator
“γ
l
∈
(E[l]/μ
l
)(K)”,
up
to
multiplication
by
±1,
or,
alterna-
2
tively,
the
index
“j
=
1”
in
the
Θ-pilot
“{q
j
}”
of
[Alien],
§3.3,
(vii)
[cf.
v
also
the
discussion
of
Example
3.8.2,
(i),
below;
the
discussion
of
Example
3.8.4,
(vi),
below].
(vii)
The
second
key
point
of
the
argument
reviewed
in
(v)
(vii-a)
consists
of
the
computation
of
dφ
discussed
at
the
beginning
of
(v)
—
i.e.,
a
computation
that
essentially
amounts
to
the
computation
of
the
logarithmic
derivative
dlog(U
)
=
dU
U
→
l
·
dlog(U
)
of
the
isogeny
φ,
written
as
“U
→
U
l
”
in
terms
of
the
standard
multi-
plicative
coordinate
“U
”
on
G
m
[cf.
(i),
(iii)]
—
which,
in
light
of
the
ampleness
of
ω
M
ell
[cf.
(iv)],
implies
that
“ω
M
ell
|
E
≈
ω
M
ell
|
E
∗
”
[i.e.,
the
“roughly
isomorphic”
arithmetic
line
bundles
obtained
by
restrict-
ing
ω
M
ell
to
E,
E
∗
]
serves,
up
to
a
negligible
discrepancy,
as
a
common
80
SHINICHI
MOCHIZUKI
container
for
the
moduli
of
both
E
and
E
∗
,
i.e.,
in
light
of
the
existence
of
the
discriminant
modular
form
“Δ
M
ell
”
[cf.
(iv)],
as
a
common
l
container
for
both
“q
E
”
and
“q
E
”.
Here,
we
observe
that
this
“common
container”/ampleness
aspect
of
ω
M
ell
may
be
understood
as
corresponding
—
cf.
the
analogy
between
étale-like
objects
in
inter-universal
Teichmüller
theory
and
tangent
bundles/sheaves
of
differen-
tials
that
was
recalled
above
in
the
context
of
(InvHt),
(FrDff)!
—
in
inter-universal
Teichmüller
theory,
to
(vii-b)
the
theory
of
the
log-link/log-shells
and
closely
related
mono-anabelian
reconstruction
algorithms
in
a
vertical
line
of
the
log-theta-lattice
that
give
rise
to
the
log-Kummer-correspondence
of
inter-universal
Teichmüller
theory,
i.e.,
which
play
the
fundamental
role
of
furnishing
a
multiradial
container
for
the
Frobenius-like
Θ-pilot
at
the
lattice
point
(0,
0)
of
the
log-theta-lattice
[cf.
the
discussion
of
§3.6,
§3.9,
§3.10,
§3.11,
below!].
Finally,
in
passing,
we
note
that
the
discriminant
modular
form
“Δ
M
ell
”
is
also
reminiscent
of
the
classical
complex
theta
function
of
Example
3.3.2
below
[i.e.,
which
may
also
be
regarded
as
a
modular
form
on
the
upper
half-plane],
as
well
as
of
the
discussion
of
the
relationship
between
the
discriminant
modular
form
and
scheme-theoretic
Hodge-Arakelov
theory
in
the
final
portion
of
[HASurI],
§1.2
[cf.
also
[Alien],
Example
2.14.3;
[Alien],
§3.9,
(i),
(ii),
for
a
discussion
of
the
relationship
between
scheme-theoretic
Hodge-Arakelov
theory
and
inter-universal
Teichmüller
theory].
(viii)
Thus,
one
may
summarize
the
discussion
of
the
present
Example
3.2.1
as
follows:
At
a
very
rough,
introductory/expository
level,
one
may
think
of
inter-
universal
Teichmüller
theory
as
a
sort
of
generalization
of
the
ap-
proach
of
(InvHt)
[cf.
(v)]
to
bounding
heights
of
elliptic
curves
over
number
fields
to
the
case
of
[essentially]
arbitrary
elliptic
curves
over
number
fields
[i.e.,
which
are
not
assumed
to
admit
a
GMS!]
by
·
somehow
“simulating”
a
GMS/GCG
and
·
applying
the
theory
of
theta
functions
and
mono-anabelian
geometry.
Example
3.2.2:
Coricity,
symmetry,
and
commutativity
properties
of
the
log-theta-lattice.
In
the
following
discussion,
we
fix
notation
as
follows:
Let
k
be
a
finite
extension
of
Q
p
,
for
some
prime
number
p;
k
an
algebraic
closure
of
k;
q
∈
k
a
nonzero
element
of
the
maximal
ideal
m
k
of
the
ring
of
integers
O
k
of
k;
N
≥
2
an
integer.
Write
·
N
for
the
additive
monoid
of
nonnegative
integers;
def
·
G
k
=
Gal(k/k);
·
O
k
for
the
ring
of
integers
of
k,
with
maximal
ideal
m
k
⊆
O
k
;
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
81
·
O
⊆
O
k
for
the
multiplicative
monoid
of
nonzero
elements
of
O
k
;
k
·
O
×
⊆
O
for
the
group
of
invertible
elements
of
O
;
k
k
k
·
O
×
O
×μ
,
k
k
×μ
×
k
k
μ
×μ
,
k
k
μ
for
the
respective
quotients
of
O
×
,
k
k
,
k
by
the
action
of
the
group
μ
∞
of
torsion
elements
[i.e.,
roots
of
unity]
of
O
×
;
k
·
F
n
=
O
×μ
×
(q
n
)
N
⊆
k
def
×μ
k
for
the
multiplicative
submonoid
of
k
×μ
[equipped
with
a
natural
action
by
G
k
]
generated
by
O
×μ
and
q
n
,
where
k
we
allow
n
to
be
an
arbitrary
positive
integer,
and,
by
a
slight
abuse
of
×μ
notation,
we
write
“q”
for
the
image
of
q
∈
m
k
in
k
;
∼
·
Θ
k
:
F
N
→
F
1
for
the
isomorphism
of
G
k
-monoids
that
restricts
to
the
identity
isomorphism
on
O
×μ
and
maps
q
N
→
q;
k
·
log
k
:
O
×
k
for
the
p-adic
logarithm
on
O
×
.
k
k
Thus,
G
k
acts
naturally
on
O
×μ
O
×
⊆
O
⊆
O
k
.
Let
Π
G
k
be
a
topological
k
k
k
group
equipped
with
a
surjection
onto
G
k
,
which
determines
natural
actions
of
Π
on
O
×μ
O
×
⊆
O
⊆
O
k
.
k
k
k
(i)
We
begin
by
observing
the
following
properties:
(i-a)
The
isomorphism
∼
Θ
k
:
F
N
→
F
1
is
not
compatible
with
the
ring
structures
in
its
domain/codomain
in
the
sense
that
it
does
not
arise
from
a
G
k
-equivariant
ring
homomorphism
φ
:
k
→
k,
i.e.,
there
does
not
exist
a
G
k
-equivariant
ring
homomorphism
×μ
×μ
φ
:
k
→
k
for
which
the
induced
map
k
→
k
restricts
either
to
a
map
F
N
→
F
1
that
coincides
with
Θ
k
or
to
a
map
F
1
→
F
N
that
coincides
with
Θ
−1
.
k
(i-b)
The
map
log
k
:
O
×
→
k
k
is
not
compatible
with
the
ring
structures
in
its
domain/codomain
in
the
sense
that
it
does
not
arise
from
a
G
k
-equivariant
ring
homomorphism
φ
:
k
→
k,
i.e.,
there
does
not
exist
a
G
k
-equivariant
ring
homomorphism
φ
:
k
→
k
that
restricts
to
log
k
.
Indeed,
both
(i-a)
and
(i-b)
follow
immediately
from
the
easily
verified
elementary
fact
that
any
G
k
-equivariant
ring
homomorphism
φ
:
k
→
k
induces
[by
passing
to
G
k
-invariants
and
considering
the
l-divisibility
properties
of
units/non-units
of
∼
k,
for
prime
numbers
l
=
p]
an
isomorphism
of
topological
fields
k
→
k,
hence
an
isomorphism
between
the
value
groups
of
the
copies
of
k
in
the
domain/codomain.
[Here,
we
recall
that,
since
N
≥
2,
the
assignments
q
N
→
q
[cf.
(i-a)]
and
(O
×
)
1
+
p
2
→
log
k
(1
+
p
2
)
(∈
p
·
O
k
)
k
[cf.
(i-b)]
82
SHINICHI
MOCHIZUKI
yield
immediate
contradictions
to
the
existence
of
such
an
induced
isomorphism
between
value
groups.]
By
contrast,
we
observe
that
∼
(i-c)
the
isomorphism
Θ
k
:
F
N
→
F
1
is
compatible
—
in
the
sense
of
equivariance,
relative
to
the
natural
actions
on
the
domain/codomain
of
∼
Θ
k
—
with
an
isomorphism
G
k
→
G
k
between
the
copies
of
G
k
in
the
domain/codomain
of
Θ
k
;
(i-d)
the
map
log
k
:
O
×
→
k
is
compatible
—
in
the
sense
of
equivariance,
k
relative
to
the
natural
actions
on
the
domain/codomain
of
log
k
—
with
an
∼
isomorphism
Π
→
Π
between
the
copies
of
Π
in
the
domain/codomain
of
log
k
.
Indeed,
(i-c)
and
(i-d)
follow
immediately
from
the
various
definitions
involved.
Here,
we
recall,
however,
that
it
is
of
fundamental
importance
to
observe
the
fol-
lowing:
(i-e)
Since
Θ
k
and
log
k
are
not
compatible
with
the
respective
ring
struc-
tures
in
their
domains/codomains
[cf.
(i-a),
(i-b)!],
in
order
to
obtain
coric
structures
—
i.e.,
structures
that
are
commonly
shared,
in
an
in-
variant
fashion,
by
these
domains/codomains,
hence
well-defined
in
a
sense
that
is
independent
of
any
specification
of
a
relationship
to
these
domains/codomains
—
it
is
necessary
to
regard
·
the
isomorphisms
∼
∼
G
k
→
G
k
and
Π
→
Π
as
indeterminate
isomorphisms
of
abstract
groups,
i.e.,
not
of
Galois
groups,
that
is
to
say,
groups
equipped
with
the
“Galois-rigidification”
constituted
by
the
auxiliary
data
of
some
sort
of
action
on
a
field/ring;
·
the
“identity
isomorphism”
given
by
restricting
Θ
k
to
O
×μ
k
∼
O
×μ
→
O
×μ
k
k
as
an
indeterminate
isomorphism
of
topological
monoids
equipped
with
an
action
by
a
topological
group
[i.e.,
G
k
],
as
well
as
with
the
collection
of
submonoids
given
by
the
images
H
in
O
×μ
of
the
intersections
O
×
∩
k
[i.e.,
of
O
×
with
the
H-
k
invariants
k
of
G
k
H
k
k
⊆
k
of
k],
where
H
ranges
over
the
open
subgroups
—
cf.
the
discussion
of
§3.8
below.
These
coricity
properties
will
also
play
an
important
role
in
the
context
of
the
discussion
of
(ii-c)
below.
In
the
context
of
(i-e),
it
is
also
important
to
note
the
following:
(i-f)
The
use
of
the
terminology
“identity
isomorphism”
when
referring
to
any
of
the
isomorphisms
∼
G
k
→
G
k
,
∼
Π
→
Π,
∼
O
×μ
→
O
×μ
k
k
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
83
discussed
in
(i-e)
can
be
highly
misleading
and
give
rise
to
unnecessary
confusion,
for
the
following
reason:
Strictly
speaking,
throughout
mathe-
matics,
this
terminology
“identity
isomorphism”
is
only
well-defined
when
applied
to
an
isomorphism
from
a
mathematical
object
to
itself
[i.e.,
not
to
a
distinct
mathematical
object!].
Since,
however,
the
ring
structures
in
the
domain/codomain
of
Θ
k
or
log
k
must
be
distinguished
[i.e.,
so
long
as
they
are
related
to
one
another
via
Θ
k
or
log
k
—
cf.
(i-a),
(i-b),
as
well
as
the
discussion
of
§3.4
below!],
the
only
possible
“well-defined
sense”
in
which
this
terminology
“identity
isomorphism”
may
be
applied
is
the
sense
of
referring
to
some
sort
of
“identity
isomorphism”
between
weaker
underlying
structures
[i.e.,
such
as
“sets”,
“abstract
topological
groups”,
“abstract
topological
monoids”,
etc.
—
cf.
the
discussion
of
§3.8
below]
that
do
indeed
coincide,
hence
may
indeed
be
“identified”
with
one
another
[cf.
the
situations
discussed
in
Example
3.5.2
below].
The
observations
of
(i-e)
and
(i-f)
play
a
fundamental
role
in
the
essential
logical
structure
of
inter-universal
Teichmüller
theory.
(ii)
Next,
we
observe
the
following
properties:
(ii-a)
The
isomorphism
∼
Θ
k
:
F
N
→
F
1
is
not
symmetric
with
respect
to
switching
the
domain/codomain
in
the
following
sense:
there
do
not
exist
G
k
-equivariant
ring
homomor-
×μ
phisms
φ
:
k
→
k,
ψ
:
k
→
k
such
that
the
induced
maps
φ
×μ
:
k
→
×μ
×μ
×μ
×μ
k
,
ψ
:
k
→
k
fit
into
a
diagram
×μ
⊇
k
⏐
⏐
×μ
F
N
Θ
k
−→
F
1
×μ
⊆
k
⏐
⏐
×μ
φ
k
ψ
×μ
⊇
F
1
Θ
k
←−
F
N
⊆
k
×μ
that
is
commutative
on
the
portion
of
the
diagram
on
which
the
relevant
composites
are
defined.
(ii-b)
The
isomorphism
∼
log
k
:
O
×
→
k
k
is
not
symmetric
with
respect
to
switching
the
domain/codomain
in
the
following
sense:
there
do
not
exist
G
k
-equivariant
ring
homomor-
phisms
φ
:
k
→
k,
ψ
:
k
→
k
that
fit
into
a
diagram
k
⏐
⏐
φ
k
⊇
O
×
k
log
k
←−
O
×
k
log
k
−→
k
⏐
⏐
ψ
⊆
k
that
is
commutative
on
the
portion
of
the
diagram
on
which
the
relevant
composites
are
defined.
Indeed,
(ii-a)
follows
by
tracing
the
image,
at
the
level
of
value
groups,
of
the
q
N
in
the
upper
left-hand
corner
of
the
diagram,
i.e.,
which
maps
[cf.
the
discussion
of
84
SHINICHI
MOCHIZUKI
induced
isomorphisms
of
value
groups
following
(i-a),
(i-b)]
to
q
via
the
composite
2
of
the
upper
horizontal
and
right-hand
vertical
arrows,
but
to
q
N
(
=
q
N
)
via
the
composite
of
the
left-hand
vertical
and
lower
horizontal
arrows.
In
a
similar
vein,
(ii-b)
follows
by
tracing
the
image,
at
the
level
of
value
groups,
of
the
element
1
u
=
{exp(p
2
)}
p
2
∈
O
×
def
k
[where
“exp(−)”
denotes
the
well-known
formal
power
series
of
the
exponential
function,
and
the
“
p
1
2
”
denotes
a
p
2
-root
of
the
element
“{−}”]
in
the
upper
left-
hand
corner
of
the
diagram,
i.e.,
which
maps
to
0
via
the
composite
of
the
upper
horizontal,
right-hand
vertical,
and
lower
horizontal
arrows,
but
to
u
(
=
0)
via
the
left-hand
vertical
arrow.
By
contrast,
we
observe
the
following:
(ii-c)
The
topological
group
actions
surrounding
Θ
k
[cf.
(i-c)]
Π
G
k
∼
→
Π
Π
Π
Θ
k
F
N
−→
G
k
→
yield
a
diagram
G
k
∼
F
1
G
k
that
is
[manifestly!]
symmetric
with
respect
to
applying
the
operation
of
∼
reflection
of
this
last
diagram
around
the
“
→
”,
where
we
observe
that
this
symmetry
property
only
holds
if
the
coricity
properties
discussed
in
(i-e),
(i-f)
are
applied,
i.e.,
if
∼
·
the
isomorphism
of
topological
groups
G
k
→
G
k
is
regarded
as
an
indeterminate
isomorphism
of
abstract
topological
groups
and
·
the
arrows
“”/“”
are
regarded
as
surjections
between
abstract
topological
groups,
i.e.,
topological
groups
that
are
only
well-defined
up
to
indeterminate
isomorphism.
Indeed,
if
one
does
not
apply
these
coricity
properties,
then
one
is
in
effect
working
with
structures
[i.e.,
the
various
“Π”
and
“G
k
”
in
“Π
∼
G
k
→
G
k
Π”]
that
depend
on
Galois-rigidifications
that
arise
from
structures
specific
to
the
domain
or
codomain
of
Θ
k
,
hence
do
not
satisfy
the
desired
symmetry
property
[cf.
the
discussion
of
(i-e),
(i-f)!].
The
observation
(ii-c),
together
with
the
observations
(i-e)
and
(i-f),
plays
a
funda-
mental
role
in
the
essential
logical
structure
of
inter-universal
Teichmüller
theory.
(iii)
Next,
we
observe
the
following
properties:
(iii-a)
Θ
k
does
not
commute
with
log
k
—
i.e.,
at
a
purely
formal
level,
“Θ
k
◦
log
k
=
log
k
◦
Θ
k
”
—
in
the
following
sense:
there
do
not
exist
G
k
-equivariant
ring
homomorphisms
φ
:
k
→
k,
ψ
:
k
→
k
such
that
the
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
μ
μ
O
×μ
⏐
k
⏐
log
⊆
μ
85
μ
induced
maps
φ
μ
:
k
→
k
,
ψ
μ
:
k
→
k
fit
into
a
diagram
F
N
Θ
k
−→
F
1
⊇
k
k
⏐
⏐
φ
μ
|
k
O
×μ
⏐
k
⏐
log
k
k
⏐
⏐
ψ
μ
|
k
μ
⊇
F
N
Θ
k
−→
F
1
⊆
k
k
μ
that
is
commutative
on
the
portion
of
the
diagram
on
which
the
relevant
composites
are
defined.
(iii-b)
Θ
k
is
not
invariant
with
respect
to
log
k
—
i.e.,
at
a
purely
formal
level,
“Θ
k
◦
log
k
=
Θ
k
”
—
in
the
following
sense:
there
do
not
exist
G
k
-
equivariant
ring
homomorphisms
φ
:
k
→
k,
ψ
:
k
→
k
such
that
the
μ
μ
μ
μ
induced
maps
φ
μ
:
k
→
k
,
ψ
μ
:
k
→
k
fit
into
a
diagram
O
×μ
⏐
k
⏐
log
⊆
F
N
Θ
k
−→
F
1
⊇
k
k
k
⏐
⏐
φ
μ
|
k
O
×μ
⏐
k
⏐
ι
μ
k
⏐
⏐
ψ
μ
k
μ
⊇
F
N
Θ
k
−→
F
1
⊆
k
μ
μ
—
where
ι
k
:
O
×μ
→
k
denotes
the
natural
inclusion
—
that
is
commu-
k
tative
on
the
portion
of
the
diagram
on
which
the
relevant
composites
are
defined.
Indeed,
let
m
be
a
positive
integer
such
that
p
m
·
O
k
⊆
log
k
(O
k
×
)
⊆
p
−m
·
O
k
,
where
we
write
O
k
×
=
O
k
∩
O
×
.
Then
(iii-a)
follows
by
tracing
the
image
of
the
element
def
k
u
n
=
exp(p
n·N
)
∈
O
×
def
k
[where
“exp(−)”
denotes
the
well-known
formal
power
series
of
the
exponential
function,
and
n
is
any
positive
integer
>
2m
[which
implies
that
n
<
n+(n−2m)
=
2n
−
2m
≤
n
·
N
−
2m]
such
that
p
n
∈
q
N
·
O
×
]
in
the
upper
left-hand
portion
of
k
the
diagram,
i.e.,
which
maps
[cf.
the
discussion
of
induced
isomorphisms
of
value
groups
following
(i-a),
(i-b)]
to
[the
image
in
the
value
group
of
k
of]
some
nonzero
element
∈
p
n·N
−2m
·O
k
[cf.
the
discussion
of
(i-e),
as
well
as
of
Example
3.5.1
below]
via
the
composite
of
the
upper
horizontal
and
right-hand
vertical
arrows,
but
to
[the
image
in
the
value
group
of
k
of]
p
n
(∈
p
n·N
−2m
·
O
k
)
via
the
left-hand
vertical
and
lower
horizontal
arrows.
In
a
similar
vein,
(iii-b)
follows
by
tracing
the
image
of
the
same
element
u
n
∈
O
×
in
the
upper
left-hand
portion
of
the
diagram,
i.e.,
which
k
maps
[cf.
the
discussion
of
induced
isomorphisms
of
value
groups
following
(i-a),
(i-b)]
to
[the
image
in
the
value
group
of
k
of]
u
n
via
the
composite
of
the
upper
horizontal
and
right-hand
vertical
arrows,
but
to
[the
image
in
the
value
group
of
k
of]
p
n
(
=
u
n
)
via
the
left-hand
vertical
and
lower
horizontal
arrows.
86
SHINICHI
MOCHIZUKI
(iv)
Finally,
we
observe
that
it
is
now
an
essentially
formal/routine
matter
to
translate
the
various
elementary
properties
discussed
above
in
(i),
(ii),
(iii)
into
the
corresponding
coricity/symmetry/commutativity
properties
of
the
log-theta-
lattice,
i.e.:
·
the
incompatibility
with
the
ring
structures
in
the
domain/codomain
[cf.
the
discussion
at
the
beginning
of
the
present
§3.2;
the
discussion
at
the
beginning
of
§3.8
below]
of
the
Θ-link
[cf.
(i-a)]
and
log-link
[cf.
(i-b)];
·
the
horizontal
[cf.
(i-c)]
and
vertical
[cf.
(i-d)]
coricity
[cf.
(i-
e),
(i-f)]
properties
of
the
étale-like
structures
that
appear
in
the
log-
theta-lattice
[cf.
the
discussion
at
the
beginning
of
the
present
§3.2;
the
discussion
at
the
beginning
of
§3.8
below];
·
the
non-symmetricity
with
respect
to
switching
the
domain/codomain
[cf.
the
discussion
at
the
beginning
of
the
present
§3.2]
of
the
Θ-link
[cf.
(ii-a)]
and
log-link
[cf.
(ii-b)];
·
the
symmetricity
with
respect
to
the
Θ-link
[cf.
the
discussion
at
the
beginning
of
the
present
§3.2]
of
the
étale-like
structures
of
the
log-theta-lattice
[cf.
(ii-c)];
·
the
non-commutativity
of
the
log-theta-lattice
[cf.
(iii-a);
the
discus-
sion
at
the
beginning
§3.3
below];
·
the
non-invariance
of
the
Θ-link
with
respect
to
the
log-link
[cf.
(iii-
b);
the
discussion
at
the
beginning
§3.3
below].
§3.3.
RCS-redundant
copies
in
the
domain/codomain
of
the
log-link
The
Θ-link
of
inter-universal
Teichmüller
theory
is
defined,
in
the
style
of
classical
complex
Teichmüller
theory
[cf.
Example
3.3.1
below;
[IUTchI],
Remark
3.9.3],
as
a
deformation
of
the
ring
structure
in
a
(Θ
±ell
NF-)Hodge
theater
that
depends,
in
an
essential
way,
on
the
splitting
into
unit
groups
and
value
groups
of
the
various
localizations
of
the
number
field
involved.
On
the
other
hand,
the
log-link
of
inter-universal
Teichmüller
theory
[i.e.,
in
essence,
the
p-adic
logarithm
at
primes
of
the
number
field
of
residue
characteristic
p]
has
the
effect
of
juggling/rotating
these
unit
groups
and
value
groups,
e.g.,
by
mapping
units
to
non-units
[cf.,
e.g.,
the
discussion
of
[Alien],
Example
2.12.3,
(v)].
In
particular,
there
is
no
natural
way
to
relate
the
two
Θ-links
[i.e.,
the
two
horizon-
tal
arrows
in
the
following
diagram]
that
emanate
from
the
domain
and
codomain
of
the
log-link
[i.e.,
the
left-hand
vertical
arrow
in
the
following
diagram]
•
Θ
−→
⏐
log
⏐
•
Θ
−→
•
.
..
??
..
.
•
—
that
is
to
say,
there
is
no
natural
candidate
for
“??”
[i.e.,
such
as,
for
instance,
an
isomorphism
or
the
log-link
between
the
two
bullets
“•”
on
the
right-hand
side
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
87
of
the
diagram]
that
makes
the
diagram
commute.
Indeed,
it
is
an
easy
exercise
[cf.
Example
3.2.2,
(iii),
(iv);
[Alien],
§3.3,
(ii);
[Rpt2018],
§15,
(LbΘ),
(Lblog),
(LbMn)],
to
show
that
neither
of
these
candidates
for
“??”
[i.e.,
an
isomorphism
or
the
log-link]
yields
a
commutative
diagram.
Thus,
in
summary,
any
identification
of
the
domain
and
codomain
of
the
log-
link
[cf.
(RC-log)!]
yields
a
situation
in
which
the
local
splittings
into
unit
groups
and
value
groups
of
the
resulting
identified
“•’s”
are
no
longer
well-defined.
In
particular,
any
such
identification
of
the
domain
and
codomain
of
the
log-link
[cf.
(RC-log)!]
yields
a
situation
in
which
the
Θ-link
is
not
well-defined
—
i.e.,
a
situation
in
which
the
apparatus
of
inter-universal
Teichmüller
theory
completely
ceases
to
function
—
cf.
the
discussion
of
the
definition
of
the
Θ-link
in
the
latter
half
of
[Alien],
§3.3,
(ii),
as
well
as
the
discussion
of
Example
3.3.3,
(i),
below.
This
discussion
may
be
summarized,
at
a
symbolic
level,
as
follows:
definition
of
the
Θ-link
=⇒
falsity
of
(RC-log).
Next,
we
observe
[cf.
the
discussion
of
[IUTchI],
Remark
3.9.3,
(iii),
(iv)]
that
the
non-existence
of
a
solution
for
“??”
in
the
above
diagram
[i.e.,
that
makes
the
diagram
commute]
amounts,
at
a
structural
level,
to
essentially
the
same
phe-
nomenon
as
the
incompatibility
of
the
dilations
that
appear
in
classical
complex
Teichmüller
theory
with
multiplication
by
non-real
roots
of
unity
[cf.
Example
3.3.1
below].
Write
R,
C,
respectively,
for
the
topological
fields
of
real
and
complex
numbers.
Then
as
observed
in
the
discussion
of
the
latter
half
of
[Alien],
§3.3,
(ii)
[cf.,
especially,
the
discussion
surrounding
[Alien],
Fig.
3.6]:
..
.
⏐
log
⏐
..
.
⏐
log
⏐
Θ
Θ
Θ
Θ
Θ
Θ
..
.
⏐
log
⏐
..
.
⏐
log
⏐
•
⏐
log
⏐
•
⏐
log
⏐
...
−→
Θ
•
−→
•
−→
.
.
.
⏐
log
⏐
log
⏐
⏐
...
−→
Θ
•
−→
•
−→
.
.
.
⏐
log
⏐
log
⏐
⏐
...
−→
Θ
•
−→
•
−→
.
.
.
⏐
log
⏐
log
⏐
⏐
•
⏐
log
⏐
•
⏐
log
⏐
..
.
..
.
..
.
..
.
⊇
Θ
•
−→
•
⏐
log
⏐
log
⏐
⏐
(InfH)
this
structural
similarity
is
consistent
with
the
analogy
discussed
in
loc.
cit.
between
·
the
“infinite
H”
portion
of
the
log-theta-lattice
consisting
of
the
two
vertical
lines
[i.e.,
of
log-links]
on
either
side
of
a
horizontal
arrow
[i.e.,
a
Θ-link]
of
the
log-theta-lattice
and
88
SHINICHI
MOCHIZUKI
·
the
elementary
theory
surrounding
the
bijection
∼
×
C
×
\GL
+
2
(R)/C
→
[0,
1)
λ
0
0
1
→
λ−1
λ+1
—
where
λ
∈
R
≥1
;
GL
+
2
(R)
denotes
the
group
of
2
×
2
real
matrices
of
positive
determinant;
C
×
denotes
the
multiplicative
group
of
C,
which
we
regard
as
a
subgroup
of
GL
+
2
(R)
via
the
a
b
,
for
a,
b
∈
R
such
that
(a,
b)
=
(0,
0);
assignment
a
+
ib
→
−b
a
the
domain
of
the
bijection
is
the
set
of
double
cosets.
That
is
to
say,
·
the
dilation
λ
0
1
0
—
cf.
the
dilations
that
appear
in
classical
complex
Teichmüller
theory,
i.e.,
as
reviewed
in
Example
3.3.1
below
—
corresponds
to
the
Θ-link
portion
of
an
“infinite
H”
[cf.
Example
3.3.2,
(iii),
below],
while
·
the
two
copies
of
the
group
of
toral
rotations
“C
×
”
[e.g.,
by
roots
of
unity
in
C
×
]
on
either
side
of
“GL
+
2
(R)”
—
which
may
be
thought
of
as
a
representation
of
the
holomorphic
structures
in
the
domain
and
codomain
of
the
dilation
λ
0
1
0
[cf.
the
discussion
of
Example
3.3.1
below]
—
correspond,
respectively,
to
the
two
vertical
lines
of
log-links
in
the
“infinite
H”
on
either
side
of
the
Θ-link
[cf.
the
discussion
of
Example
3.3.2,
(iv),
below].
Example
3.3.1:
Classical
complex
Teichmüller
theory.
Let
λ
∈
R
>1
.
Re-
call
the
most
fundamental
deformation
of
complex
structure
in
classical
complex
Teichmüller
theory
Λ:
C
C
z
=
x
+
iy
→
C
→
def
ζ
=
ξ
+
iη
=
λ
·
x
+
iy
∈
C
—
where
x,
y
∈
R.
Let
n
≥
2
be
an
integer,
ω
a
primitive
n-th
root
of
unity.
Write
(ω
∈)
μ
n
⊆
C
for
the
group
of
n-th
roots
of
unity.
Then
observe
that
if
n
≥
3,
then
there
does
not
exist
ω
∈
μ
n
such
that
Λ(ω
·
z)
=
ω
·
Λ(z)
for
all
z
∈
C.
[Indeed,
this
observation
follows
immediately
from
the
fact
that
if
n
≥
3,
then
ω
∈
R.]
That
is
to
say,
in
words,
Λ
is
not
compatible
with
multiplication
by
μ
n
unless
n
=
2
[in
which
case
ω
=
−1].
This
incompatibility
with
“indeterminacies”
arising
from
multiplication
by
μ
n
,
for
n
≥
3,
may
be
understood
as
one
fundamental
reason
for
the
special
role
played
by
square
differentials
[i.e.,
as
opposed
to
n-th
power
differentials,
for
n
≥
3]
in
classical
complex
Teichmüller
theory
[cf.
the
discussion
of
[IUTchI],
Remark
3.9.3,
(iii),
(iv)].
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
89
Example
3.3.2:
The
Jacobi
identity
for
the
classical
theta
function.
(i)
Write
z
=
x
+
iy
for
the
standard
coordinate
on
the
upper
half-plane
def
H
=
{z
=
x
+
iy
∈
C
|
y
>
0}.
Recall
the
theta
function
on
H
+∞
def
Θ(q)
=
1
2
q
2
n
.
n=−∞
def
—
where
we
write
q
=
e
2πiz
.
Restricting
to
the
imaginary
axis
[i.e.,
x
=
0]
yields
a
function
+∞
2
def
θ(t)
=
e
−πn
t
.
n=−∞
def
—
where
we
write
t
=
y.
(ii)
Next,
let
us
observe
that
01
def
ι
=
∈
C
×
⊆
GL
+
(R)
−1
0
maps
z
→
−z
−1
,
hence
iy
→
iy
−1
,
i.e.,
t
→
t
−1
,
while,
for
λ
∈
R
≥1
,
λ
0
∈
C
×
⊆
GL
+
(R)
01
maps
z
→
λ
·
z,
hence
iy
→
iλ
·
y,
i.e.,
t
→
λ
·
t.
(iii)
Next,
we
observe
the
following:
·
As
t
→
+∞,
the
terms
in
the
series
for
θ(t)
are
rapidly
decreasing,
and
θ(t)
→
+0.
In
particular,
the
series
for
θ(t)
is
relatively
easy
to
compute.
·
As
t
→
+0,
the
terms
in
the
series
for
θ(t)
decrease
very
slowly,
and
θ(t)
→
+∞.
In
particular,
the
series
for
θ(t)
is
very
difficult
to
compute.
Thus,
in
summary,
the
“flow/dilation”
λ
0
1
0
along
the
imaginary
axis
may
be
re-
garded
as
a
sort
of
“link”,
in
the
context
of
the
theta
function
θ(t),
between
small
values
[i.e.,
θ(t)
→
+0
as
t
→
+∞]
and
large
values
[i.e.,
θ(t)
→
+∞
as
t
→
+0].
That
is
to
say,
this
flow/dilation
along
the
imaginary
axis
behaves
in
a
way
that
is
strongly
reminiscent
of
the
Θ-link
of
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
(InfH)].
(iv)
The
Jacobi
identity
for
the
theta
function
θ(t)
θ(t)
=
t
−
2
·
θ(t
−1
)
1
allows
one
to
analyze
the
behavior
of
θ(t)
as
t
→
+0,
which
is
very
difficult
to
compute
[cf.
(iii)],
in
terms
of
the
behavior
of
θ(t)
as
t
→
+∞,
which
is
relatively
easy
to
compute
[cf.
(iii)]
—
cf.
the
discussion
of
the
Jacobi
identity
in
[Pano],
§3,
90
SHINICHI
MOCHIZUKI
§4;
[Alien],
§4.1,
(i).
Observe
that
this
identity
may
be
understood
as
a
sort
of
invariance
with
respect
to
ι
[cf.
(ii)],
up
to
a
certain
easily
computed
factor
[i.e.,
1
t
−
2
].
Note
that
ι
“juggles”,
or
“rotates/permutes”,
the
two
dimensions
of
R
2
.
This
aspect
of
ι
is
strongly
reminiscent
of
the
log-link
of
inter-universal
Teichmüller
theory,
which
“juggles”,
or
“rotates/permutes”,
the
two
underlying
dimensions
of
the
ring
structures
in
a
vertical
column
of
the
log-theta-lattice
[cf.,
e.g.,
the
discus-
sion
of
[Alien],
Example
2.12.3,
(v)].
By
contrast,
we
note
that
the
theta
function
θ(t)
does
not
satisfy
any
interesting
properties
of
invariance
with
respect
to
the
dilations
λ
0
1
0
.
(v)
Relative
to
the
analogy
with
the
Θ-link
and
log-link
of
inter-universal
Teichmüller
theory
discussed
in
(iii),
(iv),
the
ι-invariance
interpretation
of
the
Jacobi
identity
discussed
in
(iv)
is
strongly
reminiscent
of
the
central
role
played
by
log-link
invariance
in
the
construction
of
the
multiradial
representation
in
inter-universal
Teichmüller
theory
[cf.
the
discussion
surrounding
(logORInd),
(Di/NDi)
in
§3.11
below;
the
discussion
of
the
Jacobi
identity
in
[Pano],
§3,
§4].
1
Here,
we
note
that
the
factor
t
−
2
in
the
Jacobi
identity
may
be
understood
as
corresponding,
relative
to
the
analogy
with
inter-universal
Teichmüller
theory,
to
the
indeterminacies
(Ind1),
(Ind2),
(Ind3)
acting
on
the
log-shells
in
the
multiradial
representation.
Indeed,
both
·
the
factor
t
−
2
in
the
Jacobi
identity
—
which
amounts,
in
essence,
to
the
well-known
interpretation
of
the
theta
function
θ(t)
as
a
modular
form
on
H
—
and
1
·
the
log-shells
in
the
multiradial
representation
—
cf.
the
discussion
of
the
relationship
between
scheme-theoretic
Hodge-Arakelov
theory
and
inter-universal
Teichmüller
theory
in
[Alien],
Example
2.14.3;
[Alien],
§3.9,
(i),
(ii)
—
are
closely
related
to
the
notion
of
“differentials”.
(vi)
At
a
more
technical
level,
the
crucial
ι-invariance
property
of
(iv),
(v)
may
be
understood
as
a
consequence
of
the
Fourier
transform
invariance
of
2
the
Gaussian
“e
−
”
on
the
real
line
[cf.
the
discussion
of
the
Jacobi
identity
in
[Pano],
§3,
§4].
This
Fourier
transform
invariance
in
turn
may
be
understood
as
a
consequence
of
the
quadratic
form
“
2
”
in
the
exponent
of
the
Gaussian
2
“e
−
”,
which
may
be
thought
of
as
the
first
Chern
class
of
the
ample
line
bundle
whose
section
determines,
via
the
canonical
theta
trivialization
of
the
line
bundle,
the
theta
function
under
consideration
[cf.
the
classical
theory
of
complex
theta
functions
as
exposed,
for
instance,
in
[Mumf1],
Chapter
I].
When
this
quadratic
form
“
2
”
is
multiplied
by
a
factor
t
∈
R
>0
,
application
of
the
Fourier
transform
gives
rise,
up
to
suitable
factors
[involving,
in
particular,
the
Gaussian
integral!
—
cf.
the
discussion
of
(v),
as
well
as
[Alien],
§3.8],
to
the
transformation
e
−t·
2
e
−t
−1
·
2
that
underlies
the
Jacobi
identity.
In
this
context,
it
is
of
central
importance
to
observe
that
the
transformation
“t
t
−1
”
in
the
above
display
[i.e.,
as
opposed
to
a
transformation
of
the
form
“t
c
·
t
−1
”,
for
some
c
∈
R
>0
]
is
indicative
of
—
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
91
and
indeed
in
some
sense
essentially
equivalent
to
—
an
absolute
notion
of
“1”
[i.e.,
the
unique
invariant
element
of
the
transformation
R
>0
t
→
t
−1
∈
R
>0
]
2
in
the
copy
of
the
real
numbers
that
appears
in
the
exponent
“−
2
”
of
“e
−
”.
Finally,
we
observe
that
the
fundamental
role
played
by
the
quadratic
form
“
2
”
of
the
above
discussion
in
the
proof
of
the
Fourier
transform
invariance
of
the
Gaussian
that
underlies
the
Jacobi
identity
is
strongly
reminiscent
of
the
crucial
rigidity
properties
in
the
theory
of
the
étale
theta
function
[cf.
[Alien],
§3.4,
(iii),
(iv);
the
discussion
of
the
Jacobi
identity
in
[Pano],
§3,
§4]
that
underlie
the
multiradial
representation
of
inter-universal
Teichmüller
theory:
Indeed,
these
rigidity
properties
may
be
understood
as
consequences
of
the
theta
group
symmetries,
which
also
arise,
in
essence,
from
the
étale-theoretic
version
of
the
quadratic
form
“
2
”
of
the
above
discussion
[cf.
the
discussion
at
the
end
of
[Alien],
§3.4,
(iv)].
(vii)
Before
continuing
our
discussion,
we
pause
briefly
to
make
the
following
elementary
observation:
Let
V
be
a
1-dimensional
R-vector
space.
Then
a
[topological]
ring/field
∼
structure
on
V
may
be
understood
as
a
multiplication
map
V
⊗
R
V
→
V
given
by
an
isomorphism
of
R-vector
spaces.
By
tensoring
with
the
dual
vector
space
to
V
,
one
verifies
immediately
that
such
a
multiplication
map
∼
may
be
understood
as
an
isomorphism
of
R-vector
spaces
V
→
R,
i.e.,
as
the
choice
of
a
nonzero
element
in
V
given
by
the
image
of
1
∈
R.
In
particular,
it
follows
immediately
from
this
elementary
observation
that
the
absolute
notion
of
“1
∈
R”
discussed
in
(vi)
may
be
interpreted
as
a
[topological]
ring/field
structure
on
the
copy
of
the
real
numbers
that
appears
in
the
exponent
“(−)”
of
the
Gaussian
“e
(−)
”.
This
interpretation
is
strongly
reminiscent
of
the
central
importance,
in
inter-universal
Teichmüller
theory,
of
working
with
the
first
power
of
[the
reciprocal
of
the
l-th
root
of]
the
theta
function
[cf.,
e.g.,
the
discussion
of
[Alien],
§3.4,
(iii)],
which
makes
it
possible
to
consider
the
truncated
“mod
N
”
Kummer
theory
of
the
theta
function:
indeed,
this
truncatibility
of
the
Kummer
theory
of
the
theta
function
is
closely
related
to
the
[topological]
ring/field
structure
of
the
local
fields
that
appear
in
the
context
of
the
log-link
of
inter-universal
Teichmüller
theory
[cf.
the
analogy
between
the
log-link
and
“ι”
discussed
in
(iv);
the
discussion
of
Example
3.8.4
below;
the
discussion
of
the
final
portion
of
[Alien],
§3.6,
(ii)].
(viii)
Another
important
technical
aspect
of
the
Fourier
transform
discussed
in
(vi)
is
the
factor
“e
ixy
”
[where
x,
y
∈
R]
that
appears
in
this
Fourier
transform.
Indeed,
·
this
“exponentiation
of
a
complex
unit
∈
S
1
⊆
C
×
”
to
a
power
given
by
some
indeterminate
real
number
—
i.e.,
the
real
number
that
corresponds
to
the
variable
of
integration
in
the
Fourier
ransform
—
is
strongly
reminiscent
of
·
the
(Ind2)
indeterminacy
action
on
the
local
units
“O
×μ
”
in
inter-
universal
Teichmüller
theory
—
i.e.,
which
amounts,
in
essence,
to
expo-
nentiation
of
these
local
units
to
a
power
given
by
some
indeterminate
×
element
∈
Z
92
SHINICHI
MOCHIZUKI
[cf.
the
discussion
of
the
Jacobi
identity
in
[Pano],
§3,
§4].
Moreover,
in
this
context,
it
is
of
interest
to
note
that
the
integration
that
occurs
in
the
Fourier
˙
transform
may
be
understood
as
corresponding
to
the
logical
OR/XOR
“∨/
∨”
aspect
of
the
indeterminacies
that
occur
in
inter-universal
Teichmüller
theory
—
˙
with
addition
[where
we
recall
that
cf.
the
correspondence
of
logical
XOR
“
∨”
“integration”
may
be
understood
as
a
sort
of
“topological
addition”
operation],
as
˙
discussed
in
Example
2.4.6,
(iii),
as
well
as
in
(∧(
∨)-Chn)
in
§3.10
below.
Example
3.3.3:
Theta
functions
and
multiplicative
structures.
(i)
One
fundamental
reason
for
the
central
role
played
by
theta
functions
in
the
essential
logical
structure
of
inter-universal
Teichmüller
theory
lies
in
the
fact
that
(ThMlt)
the
main
properties
of
interest
of
the
theta
functions
that
appear
in
inter-universal
Teichmüller
theory
—
most
notably,
(ZrPl)
the
well-known
description
of
the
zeroes/poles
of
theta
func-
tions
at
the
cusps
[cf.
[EtTh],
Proposition
1.4,
(i);
[IUTchIII],
Remark
2.3.3,
(vi),
(vii);
[Alien],
§3.4,
(iii)],
which
yields
—
by
applying
the
well-known
intersection
theory
of
divisors
supported
on
the
special
fiber
of
the
universal
topological
covering
of
the
Tate
curve
[cf.
the
discussion
preceding
[EtTh],
Proposition
1.1,
of
divisors
supported
on
the
special
fiber]
—
a
“divisor-theoretic”
characterization
of
these
theta
functions
[up
to
translation
by
a
deck
transformation
of
the
universal
topological
covering];
(SymTh)
the
well-known
symmetries
of
theta
functions
[cf.
[EtTh],
Proposition
1.4,
(ii)];
(GalEv)
the
Kummer-compatible
Galois
evaluation
properties
of
theta
functions
[cf.
[IUTchII],
Remark
1.12.4;
the
discussion
at
the
beginning
of
[Alien],
§3.6;
[EtTh],
Remark
1.10.4,
(i)],
which
give
rise
to
the
canonical
splittings
of
theta
monoids,
as
well
as
to
the
construction
of
the
Gaussian
monoids
[cf.
[IUTchII],
Corol-
laries
2.5,
2.6,
3.5,
3.6;
[Alien],
§3.4,
(iii);
[Alien],
§3.6,
(ii)]
—
may
be
expressed
entirely
in
terms
of
the
multiplicative
structures
of
the
various
rings
that
appear,
i.e.,
without
invoking
the
additive
struc-
tures
of
these
rings.
This
expressibility
purely
in
terms
of
multiplicative
structures
plays
an
essential
role
in
establishing
the
·
multiradial
unit
group/value
group
splittings/decouplings
[cf.
[IUTchII],
Remark
1.12.2,
(vi);
[Alien],
§3.4,
(iii)]
and
·
non-interference
properties
[cf.
[Alien],
§3.7,
(i)]
that
underlie
the
definition
of
the
Θ-link
[cf.
the
discussion
of
the
present
§3.3
pre-
ceding
(InfH)]
and
log-Kummer-correspondence
[cf.
[IUTchII],
Remark
1.12.2,
(iv);
[IUTchIII],
Remark
1.2.3].
(ii)
In
the
context
of
(ZrPl),
it
is
important
to
recall
the
central
importance
of
the
fact
that
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
93
(ZrPlOrd)
the
signed
order
[i.e.,
“+”
for
zeroes,
“−”
for
poles]
of
the
theta
function
at
each
of
the
cusps
is
precisely
one
[cf.
[IUTchIII],
Remark
2.3.3,
(vi),
(vii);
[Alien],
§3.4,
(iii)].
This
property
(ZrPlOrd)
of
the
theta
functions
that
appear
in
inter-universal
Te-
ichmüller
theory
is
closely
related
to
the
properties
discussed
in
Example
3.3.2,
(vi),
(vii),
in
the
case
of
complex
theta
functions.
In
inter-universal
Teichmüller
theory,
this
property
(ZrPlOrd)
ensures
that
the
mono-theta-theoretic
cyclo-
tomic
rigidity
algorithms
that
arise
from
the
theory
of
theta
functions
are
·
compatible
with
the
topology
of
the
tempered
fundamental
group
and,
moreover,
are
·
not
subject
to
{±1}-indeterminacies
—
properties
that
are
not
satisfied
by
the
cyclotomic
rigidity
isomorphisms
that
arise
from
the
theory
of
algebraic
rational
functions
[cf.
[IUTchIII],
Remark
2.3.3,
(vi),
(vii);
[IUTchIII],
Remark
3.11.4;
[Alien],
§3.4,
(iii),
as
well
as
the
discussion
of
Examples
3.3.2,
(vi),
(vii);
3.8.4,
(iv),
(v),
(vi),
below,
of
the
present
paper].
Here,
it
is
of
interest
to
recall
that
(MltAdd)
the
anabelian
reconstruction
algorithms
of
[AbsTopIII],
Theorem
1.9,
im-
ply
that
the
additive
structure
of
the
function
field
of
an
algebraic
curve
may
in
fact
be
reconstructed
from
the
multiplicative
structure
of
the
func-
tion
field
of
algebraic
rational
functions
on
the
curve,
equipped
with
the
[multiplicative!]
valuation
maps
and
[multiplicative!]
evaluation
maps
at
the
closed
points
of
the
curve
[cf.
“Uchida’s
Lemma”,
i.e.,
[AbsTopIII],
Proposition
1.3].
That
is
to
say,
(MltAdd)
means
that
once
one
allows
oneself
to
work
with
algebraic
rational
functions
—
i.e.,
as
opposed
to
theta
functions
—
the
issue
emphasized
in
(i)
of
expressibility
purely
in
terms
of
multiplicative
structures
in
some
sense
ceases
to
be
well-defined/non-vacuous.
By
contrast,
if
one
restricts
oneself,
as
is
indeed
the
case
in
the
discussion
of
(i),
to
considering
theta
functions
—
as
is
necessary,
in
order
to
apply
the
essential
property
(ZrPlOrd)
discussed
above!
—
then
(MltAdd)
is
no
longer
applicable,
so
there
are
no
longer
any
obstructions
to
the
“well-definedness/non-vacuousness”
of
the
notion
of
“expressibility
purely
in
terms
of
multiplicative
structures”.
Finally,
we
recall
that,
in
any
vertical
line
of
log-links
in
the
log-theta-lattice,
·
the
discrepancy
between
the
[holomorphic]
Frobenius-like
copies
of
objects
on
either
side
of
a
log-link
[cf.
(RC-log)],
as
well
as
·
the
discrepancy
between
[holomorphic]
Frobenius-like
copies
of
objects
and
[holomorphic]
étale-like
copies
of
objects
[cf.
(RC-FrÉt)],
may
be
understood
as
the
extent
to
which
the
diagram
of
arrows
that
constitutes
the
log-Kummer-correspondence
associated
to
this
vertical
line
of
log-links
fails
to
commute.
This
failure
to
commute
may
be
estimated
by
means
of
the
indeterminacy
(Ind3),
i.e.,
by
interpreting
this
failure
to
commute
as
a
sort
of
“upper
semi-
commutativity”.
This
indeterminacy
(Ind3)
is
highly
nontrivial
and,
in
particu-
lar,
gives
rise
to
the
inequality
that
appears
in
the
final
computation
of
log-volumes
94
SHINICHI
MOCHIZUKI
in
inter-universal
Teichmüller
theory
[cf.
[IUTchIII],
Corollary
3.12].
In
this
con-
text,
it
is
important
to
recall
that
the
theory
surrounding
this
indeterminacy
(Ind3)
depends,
in
an
essential
way,
on
the
absolute
anabelian
geometry
of
[AbsTopIII],
§1,
i.e.,
which
allows
one
to
reconstruct
a
hyperbolic
curve
X
over
a
number
field
or
mixed
characteristic
local
field
from
the
abstract
profinite
group
determined
by
the
étale
fundamental
group
π
1
(X)
of
the
curve.
That
is
to
say,
in
summary,
this
absolute
anabelian
geometry
allows
one
to
show
that
the
discrepancies
between
the
various
[holomorphic]
Frobenius-like
and
[holomorphic]
étale-like
copies
of
objects
in
a
vertical
line
of
log-links
[cf.
(RC-log),
(RC-FrÉt)]
in
the
log-theta-lattice
are
“bounded
by”
the
[rel-
atively
mild]
indeterminacy
(Ind3).
On
the
other
hand,
this
absolute
anabelian
geometry
most
certainly
does
not
imply
that
these
discrepancies
are
trivial/non-existent,
i.e.,
as
asserted
in
(RC-log),
(RC-
FrÉt)
—
cf.
the
discussion
of
the
falsity
of
(RC-log),
(RC-FrÉt)
in
§3.2
and
the
present
§3.3.
§3.4.
RCS-redundant
copies
in
the
domain/codomain
of
the
Θ-link
The
Θ-link
of
inter-universal
Teichmüller
theory
Θ
•
−→
•
is
defined
as
a
gluing
between
the
(Θ
±ell
NF-)Hodge
theater
“•”
in
the
domain
of
the
arrow
and
the
(Θ
±ell
NF-)Hodge
theater
“•”
in
the
codomain
of
the
arrow
along
F
×μ
-prime-strips
“∗”
that
arise
from
the
Θ-pilot
object
“Θ-plt”
in
the
domain
and
the
q-pilot
object
“q-plt”
in
the
codomain.
Here,
it
is
important
to
note
that
this
gluing
is
obtained
by
regarding
these
F
×μ
-prime-strips
“∗”
as
being
known
only
up
to
isomorphism.
This
point
of
view,
i.e.,
of
regarding
these
F
×μ
-prime-strips
“∗”
as
being
known
only
up
to
isomorphism,
is
implemented
formally
by
taking
the
gluing
to
be
the
full
poly-isomorphism
—
i.e.,
the
set
of
all
isomorphisms
—
between
the
F
×μ
-prime-strips
arising
from
the
domain
and
codomain
of
the
Θ-link.
Here,
we
recall
that
·
q-plt
essentially
amounts
to
the
arithmetic
line
bundle
determined
by
[the
ideal
generated
by]
some
2l-th
root
q
of
the
q-parameter
at
the
valuations
v
v
∈
V
bad
,
while
·
Θ-plt
essentially
amounts
to
the
collection
of
arithmetic
line
bundles
de-
2
termined
by
[the
ideals
generated
by]
the
collection
{q
j
},
as
j
ranges
over
v
def
the
integers
1,
.
.
.
,
l
=
l−1
2
[where
l
is
the
prime
number
that
appears
in
the
initial
Θ-data
under
consideration],
and
v
ranges
over
the
valuations
∈
V
bad
.
Also,
we
recall
that
each
(Θ
±ell
NF-)Hodge
theater
“•”
gives
rise
to
an
associated
model
“Ring”
of
the
ring/scheme
theory
surrounding
the
elliptic
curve
under
con-
sideration.
In
the
following
discussion,
we
shall
write
·
†
•
for
the
“•”
in
the
domain
of
the
Θ-link,
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
95
·
‡
•
for
the
“•”
in
the
codomain
of
the
Θ-link,
·
for
an
arbitrary
element
of
the
set
consisting
of
“†”,
“‡”,
and
the
“empty
symbol”
[i.e.,
no
symbol
at
all],
·
Θ-plt
∈
Ring
for
the
Θ-pilot
arising
from
the
collection
“{
q
j
}”
that
2
v
appears
in
the
model
of
ring/scheme
theory
associated
to
•,
and
·
q-plt
∈
Ring
for
the
q-pilot
arising
from
the
“
q
”
that
appears
in
the
model
of
ring/scheme
theory
associated
to
•.
v
Finally,
we
recall
that
since,
for
j
=
1,
the
valuation
[at
each
valuation
v
∈
V
bad
]
2
of
q
j
differs
from
that
of
q
,
the
arithmetic
degrees
of
the
line
bundles
constituted
v
by
q-plt
and
Θ-plt
differ.
v
Thus,
at
a
more
formal
level,
the
above
description
of
the
gluing
that
consti-
tutes
the
Θ-link
may
be
summarized
as
follows:
†
Ring
†
Θ-plt
←:
∗
:→
‡
q-plt
∈
‡
Ring
Ring
q-plt
=
Θ-plt
∈
Ring
[where
“←:”
and
“:→”
denote
the
assignments
that
consitute
the
gluing
discussed
above].
In
this
context,
we
note
the
following
fundamental
observation,
which
un-
derlies
the
entire
logical
structure
of
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
[IUTchIII],
Remark
3.12.2,
(c
itw
),
(f
itw
);
[Alien],
§3.11,
(iv)]:
(AOΘ1)
the
following
condition
holds:
†
†
‡
‡
∗
:→
Θ-plt
∈
Ring
∧
∗
:→
q-plt
∈
Ring
.
By
contrast,
if
one
simply
deletes
the
distinct
labels
“†”,
“‡”
[cf.
(RC-Θ)!],
then
(AOΘ2)
the
following
condition
holds:
∗
:→
Θ-plt
∈
Ring
∨
∗
:→
q-plt
∈
Ring
.
Of
course,
(AOΘ3)
the
essential
mathematical
content
discussed
in
this
condition
(AOΘ2)
may
be
formally
described
as
a
condition
involving
the
AND
relator
“∧”:
q-plt
∈
{q-plt,
Θ-plt}
∧
Θ-plt
∈
{q-plt,
Θ-plt}
.
On
the
other
hand,
precisely
as
a
consequence
of
the
fact
[discussed
above]
that
Ring
q-plt
=
Θ-plt
∈
Ring,
96
SHINICHI
MOCHIZUKI
(AOΘ4)
the
following
condition
does
not
hold:
∗
:→
Θ-plt
∈
Ring
∧
∗
:→
q-plt
∈
Ring
.
That
is
to
say,
the
operation
of
identifying
†
•,
‡
•
[hence
also
†
Ring,
‡
Ring]
—
e.g.,
on
the
grounds
of
“redundancy”
[i.e.,
as
asserted
in
(RC-Θ)!]
—
by
deleting
the
distinct
labels
“†”,
“‡”
has
the
effect
of
passing
from
a
situation
in
which
the
AND
relator
“∧”
holds
[cf.
(AOΘ1)]
to
a
situation
in
which
the
OR
relator
“∨”
holds
[cf.
(AOΘ2),
(AOΘ3)],
but
the
AND
relator
“∧”
does
not
hold
[cf.
(AOΘ4)]!
In
particular,
relative
to
the
correspondences
†
•,
†
Ring
←→
†
†
I;
Θ-plt
∗
←→
†
←→
‡
β;
‡
γ
J
;
q-plt
•,
‡
Ring
←→
‡
←→
‡
I
α
[cf.
the
correspondences
(StR1)
∼
(StR6)
discussed
in
§3.2;
the
correspondences
discussed
in
Example
2.4.5,
(ii);
the
discussion
of
[Alien],
§3.11,
(iv)],
one
obtains
very
precise
structural
resemblances
(AOΘ1)
(AOΘ2)
(AOΘ3)
(AOΘ4)
←→
←→
←→
←→
(AOL1),
(AOL2),
(AOL3),
(AOL4)
with
the
situation
discussed
in
Example
2.4.1,
(i),
(ii).
Thus,
in
summary,
the
falsity
of
(RC-Θ)
may
be
understood
as
a
consequence
of
the
falsity
[cf.
(AOΘ4)]
of
the
crucial
AND
relator
“∧”
in
the
absence
of
distinct
labels,
in
stark
contrast
to
the
truth
[cf.
(AOΘ1)]
of
the
crucial
AND
relator
“∧”
as
an
essentially
tautological
consequence
of
the
use
of
the
distinct
labels
“†”,
“‡”.
In
the
context
of
the
central
role
played
in
the
logical
structure
of
inter-universal
Teichmüller
theory
by
the
validity
of
(AOΘ1),
it
is
important
to
note
[cf.
the
property
discussed
in
(AOΘ4)!]
that
∼
(NoRng)
there
does
not
exist
an
isomorphism
of
ring
structures
†
Ring
→
‡
Ring
that
induces,
on
value
groups
of
corresponding
local
rings,
the
desired
2
assignment
{
†
q
j
}
→
‡
q
[i.e.,
that
appears
in
the
Θ-link].
v
v
On
the
other
hand,
if,
instead
of
considering
the
full
ring
structures
of
†
Ring,
‡
Ring,
one
considers
[cf.
the
discussion
of
[Rpt2018],
§6]
·
certain
suitable
subquotients
—
i.e.,
in
the
notation
of
[Alien],
§3.3,
(vii),
(a
q
),
(a
Θ
),
“O
×
”
—
of
the
underlying
multiplicative
monoids
k
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
97
of
corresponding
local
fields,
as
well
as
·
the
absolute
Galois
groups
—
i.e.,
in
the
notation
of
[Alien],
§3.3,
(vii),
(a
q
),
(a
Θ
),
“G
k
”
—
associated
to
corresponding
local
rings,
regarded
as
abstract
topological
groups
[that
is
to
say,
not
as
Galois
groups,
or
equivalently/alternatively,
as
groups
of
field
automorphisms!
—
cf.
the
discussion
of
§3.8
below],
then
one
obtains
structures
—
i.e.,
the
structures
that
constitute
the
F
×μ
-prime-
strips
that
appear
in
the
Θ-link
—
that
are
simultaneously
associated
[as
“un-
derlying
structures”]
to
both
†
Ring
and
‡
Ring
via
isomorphisms
[i.e.,
of
certain
suitable
multiplicative
monoids
equipped
with
actions
by
certain
suitable
abstract
topological
groups]
that
restrict,
on
the
subquotient
monoids
that
correspond
to
the
2
respective
value
groups,
to
the
desired
assignment
{
†
q
j
}
→
‡
q
.
It
is
this
crucial
v
v
simultaneity
that
yields,
as
a
tautological
consequence,
the
validity
of
the
AND
relator
“∧”
in
(AOΘ1).
Working,
as
in
the
discussion
above,
with
multiplicative
monoids
equipped
with
actions
by
abstract
topological
groups,
necessarily
gives
rise
to
certain
inde-
terminacies,
called
(Ind1),
(Ind2),
that
play
an
important
role
in
inter-universal
Teichmüller
theory.
Certain
aspects
of
these
indeterminacies
(Ind1),
(Ind2)
will
be
discussed
in
more
detail
in
§3.5
below.
In
this
context,
we
recall
that
one
central
assertion
of
the
RCS
[cf.
the
discussion
of
Example
3.2.2;
the
discussion
of
(SSInd),
(SSId)
in
[Rpt2018],
§7,
§10]
is
to
the
effect
that
(NeuRng)
these
indeterminacies
(Ind1),
(Ind2)
may
be
eliminated,
without
affecting
the
essential
logical
structure
of
inter-universal
Teichmüller
theory,
by
tak-
ing
the
multiplicative
monoids
and
abstract
topological
groups
that
appear
in
the
F
×μ
-prime-strips
of
the
above
discussion
to
be
equipped
with
rigidifications
by
regarding
them
as
arising
from
some
fixed
“neutral”
ring
structure
Ring.
On
the
other
hand,
as
discussed
in
(NoRng)
above,
there
does
not
exist
any
ring
structure
that
is
compatible
[i.e.,
in
the
sense
discussed
in
(NoRng)],
with
the
desired
2
assignment
{
†
q
j
}
→
‡
q
.
That
is
to
say,
in
summary,
v
v
(NeuORInd)
working
with
such
a
fixed
“neutral”
ring
structure
Ring
as
in
(NeuRng)
means
either
that
(NeuORInd1)
there
is
no
relationship
between
“∗”
and
Ring,
or
that
(NeuORInd2)
the
relationship
between
“∗”
and
Ring
is
always
necessarily
subject
to
an
indeterminacy
[cf.
(AOΘ2),
(AOΘ3)!]
∗
:→
Θ-plt
∈
Ring
∨
∗
:→
q-plt
∈
Ring
[cf.
the
situation
discussed
in
Example
3.2.2;
the
situation
discussed
in
[Rpt2018],
§10,
(SSId)].
Here,
we
observe
that
whichever
of
these
“options”/“indeterminacies”
that
appear
in
(NeuORInd)
[i.e.,
(NeuORInd1),
(NeuORInd2)]
one
chooses
to
adopt,
one
is
forced
to
contend
with
an
indeterminacy
that
is,
in
some
sense,
much
more
drastic
98
SHINICHI
MOCHIZUKI
than
the
relatively
mild
indeterminacies
(Ind1),
(Ind2)
whose
elimination
formed
the
original
motivation
for
the
introduction
of
Ring!
Finally,
we
observe
that
this
much
more
drastic
indeterminacy
(NeuORInd)
means
[cf.
the
discussion
of
Example
2.4.4!]
that
throughout
any
argument,
one
must
always
take
the
position
that
the
only
possible
relationship
between
“∗”
and
Θ-plt,
q-plt
is
one
in
which
(PltRel)
“∗”
maps
either
to
Θ-plt
or
—
i.e.,
“∨”!
—
to
q-plt,
but
not
both!
Since
‡
Ring
may
be
thought
of
as
a
ring
structure
in
which
“∗”
tautologically
maps
to
‡
Θ-plt,
while
†
Ring
may
be
thought
of
as
a
ring
structure
in
which
“∗”
tauto-
logically
maps
to
‡
q-plt,
one
may
rephrase
the
above
observation
as
the
observation
that
one
must
always
take
the
position
that
the
only
possible
relationship
between
Ring,
on
the
one
hand,
and
‡
Ring,
†
Ring,
on
the
other,
is
one
in
which
(RngRel)
the
ring
structure
Ring
is
identified
either
with
the
ring
structure
‡
Ring
or
—
i.e.,
“∨”!
—
with
the
ring
structure
†
Ring,
but
not
both!
At
this
point,
let
us
recall
[cf.,
e.g.,
the
discussion
of
§3.5,
§3.11,
below;
[Rpt2018],
§9,
(GIUT),
(ΘCR)]
that
inter-universal
Teichmüller
theory
requires,
in
an
essential
way,
the
use
of
the
log-links,
hence,
in
particular,
[in
order
to
define
the
power
series
of
the
various
p-adic
logarithm
functions
that
constitute
these
log-links!]
the
ring
structures
†
Ring,
‡
Ring
on
both
sides
—
i.e.,
“∧”!
—
of
the
Θ-link
[cf.
the
discussion
surrounding
(InfH)
of
the
two
vertical
lines
of
log-links
in
the
“infinite
H”
on
either
side
of
the
Θ-link].
In
particular,
we
conclude
formally
that
it
is
impossible
to
implement
the
arguments
of
inter-universal
Teichmüller
theory
once
this
sort
of
much
more
drastic
indeterminacy
(NeuORInd)
has
been
imposed.
§3.5.
Gluings,
indeterminacies,
and
pilot
discrepancy
As
discussed
in
§3.4,
the
Θ-link
involves
a
gluing
{
†
q
j
}
→
‡
q
2
v
v
that
identifies
‡
q
[i.e.,
2l-th
roots
of
the
q-parameters
at
primes
of
multiplicative
v
reduction
of
the
[copy
belonging
to
‡
Ring
of
the]
elliptic
curve
under
consideration]
2
with
elements,
i.e.,
the
†
q
j
’s,
which,
when
j
=
1,
have
different
valuations
from
the
valuation
of
‡
q
.
v
v
On
the
other
hand,
in
inter-universal
Teichmüller
theory,
by
applying
the
mul-
tiradial
representation
of
[IUTchIII],
Theorem
3.11,
which
involves
various
in-
determinacies
(Ind1),
(Ind2),
(Ind3),
and
then
forming
[cf.
[IUTchIII],
Corollary
3.12,
and
its
proof]
the
holomorphic
hull
of
the
union
of
possible
images
of
the
Θ-pilot
in
this
multiradial
representation,
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
99
(ΘGl)
one
may
treat
both
sides
of
the
Θ-link
gluing
of
the
above
display
as
belonging
to
a
single
ring
theory
without
disturbing
[cf.
the
crucial
AND
relator
“∧”
property
discussed
in
§3.4!]
the
gluing.
Alternative
ways
to
understand
the
essential
mathematical
content
of
(ΘGl)
include
the
following:
(NonInf)
One
may
think
of
(ΘGl)
as
a
statement
concerning
the
mutually
non-
interference
or
simultaneous
executatibility
of
the
Kummer
theo-
ries
surrounding
the
q-pilot
and
Θ-pilot
relative
to
the
gluing
of
abstract
F
×μ
-prime-strips
constituted
by
the
Θ-link,
i.e.,
when
the
Kummer
theory
surrounding
the
q-pilot
is
held
fixed,
and
one
allows
the
Kummer
theory
surrounding
the
Θ-pilot
to
be
subject
to
various
indeterminacies.
(Cohab)
One
may
think
of
(ΘGl)
as
a
statement
concerning
the
“cohabitation”,
or
“coexistence”,
of
the
q-pilot
and
Θ-pilot
—
relative
to
the
gluing
of
abstract
F
×μ
-prime-strips
constituted
by
the
Θ-link
—
within
the
common
container
obtained
by
applying
the
multiradial
representation
of
the
Θ-pilot,
forming
the
holomorphic
hull
[relative
to
the
holomorphic
structure
[i.e.,
(Θ
±ell
NF-)Hodge
theater]
that
gave
rise
to
the
q-pilot
under
consideration],
and
finally
taking
log-volumes.
In
this
context,
it
is
important
to
recall
that
this
sort
of
phenomenon
—
i.e.,
of
computations
of
global
degrees/heights
of
elliptic
curves
in
sit-
uations
where
a
certain
“confusion”,
up
to
suitable
indeterminacies,
is
allowed
between
q-parameters
of
the
elliptic
curves
and
certain
large
positive
powers
of
these
q-parameters
[i.e.,
as
in
(ΘGl)]
—
may
be
seen
in
various
classical
examples
such
as
·
the
proof
by
Faltings
of
the
invariance
of
heights
of
abelian
varieties
under
isogeny
[cf.
the
discussion
of
[Alien],
§2.3,
§2.4,
as
well
as
the
discussion
of
Example
3.2.1
in
the
present
paper],
·
the
classical
proof
in
characteristic
zero
of
the
geometric
version
of
the
Szpiro
inequality
via
the
Kodaira-Spencer
morphism,
phrased
in
terms
of
the
theory
of
crystals
[cf.
the
discussion
of
[Alien],
§3.1,
(v)],
and
·
Bogomolov’s
proof
over
the
complex
numbers
of
the
geometric
version
of
the
Szpiro
inequality
[cf.
the
discussion
of
[Alien],
§3.10,
(vi)]
—
cf.
also
the
discussion
of
[Rpt2018],
§16.
Moreover,
in
the
case
of
crystals,
we
observe
that,
relative
to
the
notation
introduced
in
Example
2.4.5,
(v),
(vi),
we
have
correspondences
as
follows:
(CrAND)
The
logical
AND
“∧”
that
appears
in
the
multiradial
representation
of
the
Θ-pilot
in
IUT
(=
AND-IUT)
may
be
understood
as
being
analogous
to
the
fact
that
crystals,
i.e.,
“∧-crystals”,
may
be
thought
of
as
objects
[on
infinitesimal
neighborhoods
of
the
diagonal
inside
products
of
two
copies
of
the
scheme
under
consideration]
that
may
be
simultaneously
interpreted,
up
to
isomorphism,
as
pull-backs
via
one
projection
morphism
and
[cf.
“∧”!]
as
pull-backs
via
the
other
projection
morphism
[cf.
the
˙
discussion
of
(∧(
∨)-Chn1)
in
§3.10
below;
the
discussion
of
[Alien],
§3.11,
100
SHINICHI
MOCHIZUKI
(iv),
(2
and
),
concerning
the
interpretation
of
the
discussion
of
crystals
in
[Alien],
§3.1,
(v),
(3
KS
),
in
terms
of
the
logical
relator
“∧”].
(CrOR)
Thus,
from
the
point
of
view
of
the
analogy
discussed
in
(CrAND),
the
logical
OR
“∨”
that
appears
throughout
OR-IUT
may
be
understood
as
corresponding
to
working
with
“∨-crystals”
[i.e.,
as
opposed
to
crystals
(=
∧-crystals)],
that
is
to
say,
with
objects
[on
infinitesimal
neighbor-
hoods
of
the
diagonal
inside
products
of
two
copies
of
the
scheme
under
consideration]
that
may
be
interpreted,
up
to
isomorphism,
as
pull-backs
via
one
projection
morphism
or
[cf.
“∨”!]
as
pull-backs
via
the
other
pro-
jection
morphism.
Here,
we
observe
that
this
defining
“∨”
condition
of
an
∨-crystal
is
essentially
vacuous
since
one
may
obtain
∨-crystals
from
ar-
bitrary
objects
on
the
scheme
under
consideration
simply
by
pulling
back
such
an
object
to
the
infinitesimal
neighborhood
of
the
diagonal
under
consideration
via
one
of
the
two
projection
morphisms.
(CrRCS)
In
a
similar
vein,
from
the
point
of
view
of
the
analogies
discussed
in
(CrAND)
and
(CrOR),
RCS-IUT
may
be
understood
as
corresponding
to
the
modified
version
of
the
usual
theory
of
crystals
obtained
by
re-
placing
the
infinitesimal
neighborhoods
of
the
diagonal
inside
products
of
two
copies
of
the
scheme
under
consideration
[i.e.,
that
appear
in
the
usual
theory
of
crystals!]
by
the
diagonal
itself.
Such
a
replacement
clearly
renders
the
usual
theory
of
crystals
trivial/meaningless,
in
a
fashion
that
is
essentially
very
similar
to
the
triviality
of
∨-crystals
dis-
cussed
in
(CrOR).
Finally,
we
observe
that
this
similarity
between
the
modified
versions
of
the
usual
theory
of
crystals
discussed
in
(CrOR)
and
the
present
(CrRCS)
is
entirely
analogous
to
the
equivalence
OR-IUT
⇐⇒
RCS-IUT
observed
in
Example
2.4.5,
(v),
(XOR/RCS).
Unfortunately,
however,
the
situation
summarized
above
in
(ΘGl)
has
resulted
in
certain
frequently
voiced
misunderstandings
by
some
mathematicians.
One
such
frequently
voiced
misunderstanding
is
to
the
effect
that
(CnfInd1+2)
the
situation
summarized
in
(ΘGl)
may
be
explained
as
a
consequence
of
a
“confusion”
between
q-parameters
and
large
positive
powers
of
these
q-parameters
that
results
from
the
indeterminacies
(Ind1),
(Ind2).
In
fact,
however,
as
discussed
in
Example
3.5.1,
(iii),
below,
at
least
in
the
case
of
q-parameters
of
sufficiently
small
valuation
[i.e.,
sufficiently
large
positive
order,
in
the
sense
of
loc.
cit.],
such
a
“con-
fusion”
[i.e.,
between
q-parameters
and
large
positive
powers
of
these
q-parameters]
can
never
occur
as
a
consequence
of
(Ind1),
(Ind2),
i.e.,
both
of
which
amount
to
automorphisms
of
the
[underlying
topological
module
of
the]
log-shells
involved
[cf.
also
the
discussion
of
(ΘInd)
in
[Rpt2018],
§11].
In
this
context,
we
note
that
this
misunderstanding
(CnfInd1+2)
appears
to
be
caused
in
many
cases,
at
least
in
part,
by
a
more
general
misunderstanding
concerning
the
operation
of
passage
to
underlying
structures
[cf.
Example
3.5.2
below].
A
more
detailed
discussion
of
the
operation
of
passage
to
underlying
structures
may
be
found
in
§3.9
below.
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
101
As
discussed
in
[Rpt2018],
§11,
the
“confusion”
summarized
in
(ΘGl)
occurs
in
inter-universal
Teichmüller
theory
as
a
consequence
not
only
of
the
local
in-
determinacies
(Ind1),
(Ind2),
(Ind3),
but
also
of
the
constraints
imposed
by
the
global
realified
Frobenioid
portions
of
the
F
×μ
-prime-strips
that
appear
in
the
Θ-link.
In
this
context,
it
is
of
particular
importance
to
observe
that
(CnfInd3)
the
indeterminacy
(Ind3),
which
constrains
one
to
restrict
one’s
atten-
tion
to
upper
bounds
[i.e.,
but
not
lower
bounds!]
on
the
log-volume
that
is
the
subject
of
the
computation
of
[IUTchIII],
Corollary
3.12,
al-
ready
by
itself
—
i.e.,
without
considering
(Ind1),
(Ind2),
or
global
reali-
fied
Frobenioids!
[cf.
the
discussion
of
(Ind3>1+2)
in
§3.11
below]
—
is
sufficient
to
account
for
the
possibility
of
a
“confusion”
of
the
sort
summarized
in
(ΘGl)
[i.e.,
between
q-parameters
and
large
positive
powers
of
these
q-parameters].
Indeed,
the
indeterminacy
(Ind3)
is
defined
in
precisely
such
a
way
as
to
identify
the
ideals
generated
by
arbitrary
positive
powers
of
the
q-parameters.
Example
3.5.1:
Bounded
nature
of
log-shell
automorphism
indetermina-
cies.
Write
Z
p
for
the
ring
of
p-adic
integers,
for
some
prime
number
p;
Q
p
for
the
field
of
fractions
of
Z
p
.
(i)
Let
M
be
a
finitely
generated
free
Z
p
-module,
which,
in
the
following
dis-
def
cussion,
we
shall
think
of
as
being
embedded
in
M
Q
p
=
M
⊗
Z
p
Q
p
;
∼
α
:
M
→
M
an
automorphism
of
the
Z
p
-module
M
.
For
n
∈
Z,
write
def
U
(M,
n)
=
{x
∈
M
Q
p
|
x
∈
p
n
·
M,
x
∈
p
n+1
·
M
}
⊆
M
Q
p
.
Then
observe
that
α
induces
a
bijection
∼
U(M,
n)
→
U(M,
n)
for
every
n
∈
Z.
(ii)
In
the
notation
of
(i),
suppose,
for
simplicity,
that
p
is
odd.
Let
k
be
a
finite
field
extension
of
Q
p
.
Write
O
k
⊆
k
for
the
ring
of
integers
of
k;
O
k
×
⊆
O
k
for
the
group
of
units
of
O
k
;
m
k
⊆
O
k
for
the
maximal
ideal
of
k;
I
k
⊆
k
for
the
log-shell
associated
to
k
[cf.,
e.g.,
the
discussion
of
[IUTchIII],
Remark
1.2.2,
(i)],
i.e.,
the
result
of
multiplying
by
p
−1
the
image
log
p
(O
k
×
)
of
O
k
×
by
the
p-adic
logarithm
log
p
(−).
Thus,
O
k
⊆
I
k
⊆
p
−c
·
O
k
for
some
nonnegative
integer
of
c
that
depends
only
on
the
isomorphism
class
of
the
field
k
[cf.
[IUTchIV],
Proposition
1.2,
(i)].
In
particular,
there
exists
a
positive
integer
s
that
depends
only
on
the
isomorphism
class
of
the
field
k
such
that
for
any
automorphism
∼
φ
:
I
k
→
I
k
102
SHINICHI
MOCHIZUKI
of
the
Z
p
-module
I
k
and
any
n
∈
Z,
it
holds
that
φ(U
(O
k
,
n))
⊆
s
U(O
k
,
n
+
i)
i=−s
[where
i
ranges
over
the
integers
between
−s
and
s].
(iii)
In
the
situation
of
(ii),
we
define
the
order
of
a
nonzero
element
x
∈
k
to
be
.
One
thus
concludes
from
the
final
the
unique
n
∈
Z
such
that
x
∈
m
nk
,
x
∈
m
n+1
k
portion
of
the
discussion
of
(ii)
that
there
exists
a
positive
integer
t
that
depends
only
on
the
isomorphism
class
of
the
field
k
such
that
for
any
automorphism
∼
φ
:
I
k
→
I
k
of
the
Z
p
-module
I
k
and
any
nonzero
element
q
∈
O
k
[i.e.,
such
as
the
q-parameter
of
a
Tate
curve
over
k!],
the
absolute
value
of
the
difference
between
the
orders
of
q
and
φ(q)
is
≤
t,
i.e.,
in
words,
automorphisms
of
the
Z
p
-module
I
k
only
give
rise
to
bounded
discrep-
ancies
in
the
orders
of
nonzero
elements
of
O
k
.
Example
3.5.2:
Examples
of
gluings.
Distinct
auxiliary
structures
on
some
common
[i.e.,
“∧”!]
underlying
structure
may
be
thought
of
as
gluings
of
the
dis-
tinct
auxiliary
structures
along
the
common
underlying
structure.
Here,
we
observe
that,
in
general,
distinct
auxiliary
structures
on
a
common
underlying
structure
are
not
necessarily
mapped
to
one
another
by
some
automorphism
of
the
common
un-
derlying
structure.
Concrete
examples
of
these
generalities
may
be
found
in
quite
substantial
abundance
throughout
arithmetic
geometry
and
include,
in
particular,
the
examples
(i),
(ii),
(iii),
(iv),
(v)
given
below,
as
well
as
the
elementary
Examples
2.3.2,
2.4.1,
2.4.2,
2.4.3,
2.4.7,
2.4.8,
3.3.1
discussed
in
§2.3,
§2.4,
§3.3
[cf.
also
the
discussion
of
[Rpt2018],
§11].
In
passing,
we
observe
that
these
examples
may
also
be
understood
as
interesting
examples
of
the
sort
of
gluing/logical
AND
“∧”
relation
that
appears
in
the
Θ-/log-
links
of
inter-universal
Teichmüller
theory,
i.e.,
examples
of
situations
that
are
qualitatively
similar
to
the
Θ-/log-links
of
inter-universal
Teichmüller
theory
in
the
sense
that
they
involve
distinct
auxiliary
structures
that
are
glued
together
along
some
common
auxiliary
structure
[cf.
the
discussion
of
(StR1)
∼
(StR6)
in
§3.2;
the
discussion
of
§3.4;
the
portion
of
the
present
§3.5
preceding
Example
3.5.1].
(i)
The
group
structures
of
the
finite
abelian
groups
Z/2Z×Z/2Z
and
Z/4Z
are
not
mapped
to
one
another
by
any
isomorphism
of
sets,
despite
the
fact
that
the
underlying
sets
of
these
two
groups
are
indeed
isomorphic
to
one
another.
This
example
is
also
of
interest
in
light
of
the
discussion
of
truncated
Witt
vectors
in
Example
2.4.6,
(iii).
(ii)
The
scheme
structures
of
non-isomorphic
algebraic
curves
over
a
com-
mon
algebraically
closed
field
are
not
mapped
to
one
another
by
any
isomorphism
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
103
of
topological
spaces,
despite
the
fact
that
the
underlying
topological
spaces
of
al-
gebraic
curves
over
a
common
algebraically
closed
field
are
indeed
isomorphic
to
one
another.
(iii)
The
holomorphic
structures
of
non-isomorphic
compact
Riemann
sur-
faces
R
1
,
R
2
with
homeomorphic
underlying
topological
spaces
are
not
mapped
to
one
another
by
any
homeomorphism,
i.e.,
by
any
isomorphism
of
topological
spaces,
despite
the
fact
that
the
underlying
topological
spaces
of
such
Riemann
surfaces
R
1
,
R
2
are
indeed
isomorphic
to
one
another.
[This
example
is
in
fact
essentially
similar
to
the
situation
discussed
in
Example
3.3.1,
except
that
in
the
situation
of
Example
3.3.1,
the
“two”
Riemann
surfaces
involved
are
both
isomorphic
to
the
complex
plane,
hence,
in
particular,
isomorphic
to
one
another.]
In
this
context,
it
is
of
interest
to
observe
that
(iii-a)
if
one
defines
the
genus
of
such
a
compact
Riemann
surface
R
i
,
for
i
∈
{1,
2},
as
the
complex
dimension
of
the
space
of
global
holomorphic
differentials
on
the
Riemann
surface,
then
it
is
by
no
means
clear
that
R
1
and
R
2
have
the
same
genus,
i.e.,
since
this
definition
of
the
genus
depends,
in
an
essential
way,
on
the
holomorphic
structure
of
the
Rie-
mann
surface.
On
the
other
hand,
once
one
verifies
that
(iii-b)
this
“holomorphic
definition”
of
the
genus
coincides
with
the
genus
de-
fined
in
terms
of
the
singular
homology
group
of
the
underlying
topo-
logical
space,
it
follows
immediately
that
R
1
and
R
2
do
indeed
have
the
same
genus.
This
contrast
between
the
“holomorphic”
and
“topological”
definitions
of
the
genus
is
of
interest
in
the
context
of
inter-universal
Teichmüller
theory
since
it
illustrates
(iii-c)
the
very
substantive
significance
of
formulating
the
definitions
of
objects
or
constructions
[i.e.,
in
the
present
discussion,
the
“genus”]
in
terms
of
structures
that
are
coric
for
the
“gluing/link”
[i.e.,
in
the
present
dis-
cussion,
a
comparison
of
distinct
holomorphic
structures
on
homeomor-
phic
topological
spaces]
under
consideration,
that
is
to
say,
in
terms
of
structures
that
are
commonly
shared
in
an
invariant
fashion
by,
hence
satisfy
a
logical
AND
“∧”
relation
relative
to,
the
two
objects
[i.e.,
in
the
present
discussion,
R
1
and
R
2
]
that
are
related
to
one
another
by
the
link
under
consideration.
We
refer
to
the
discussion
of
§3.8
below
for
a
more
detailed
treatment
of
the
im-
portance
of
coric
structures
in
inter-universal
Teichmüller
theory.
(iv)
The
field
structures
of
non-isomorphic
mixed-characteristic
local
fields
[which,
by
local
class
field
theory,
may
be
regarded
as
[the
formal
union
with
“{0}”
of]
some
suitable
subquotient
of
their
respective
absolute
Galois
groups]
are
not,
in
general,
mapped
to
one
another
by
any
isomorphism
of
profinite
groups
between
the
respective
absolute
Galois
groups
[cf.,
e.g.,
[Ymgt],
§2,
Theorem,
for
an
example
of
this
phenomenon].
(v)
In
the
notation
of
Example
3.5.1,
(i),
let
X
be
a
proper
smooth
curve
of
def
genus
≥
2
over
F
p
=
Z
p
/pZ
p
.
Thus,
X
may
be
thought
of
as
an
“underlying
104
SHINICHI
MOCHIZUKI
structure”
associated
to
any
lifting
of
X
to
Z
p
,
i.e.,
any
flat
Z
p
-scheme
Y
equipped
∼
an
isomorphism
of
F
p
-schemes
Y
×
Z
p
F
p
→
X.
Then
observe
that
non-isomorphic
liftings
of
X
to
Z
p
are
not,
in
general,
mapped
to
one
another
by
any
automorphism
of
the
F
p
-scheme
X.
[Indeed,
this
is
particularly
easy
to
see
if
one
chooses
X
such
that
X
does
not
admit
any
nontrivial
automorphisms.]
In
passing,
we
note
that
this
example
may
be
regarded
as
a
sort
of
p-adic
analogue
of
the
example
of
(iii).
Finally,
we
make
the
important
observation
that
crystals
on
X
[relative
to
Z
p
]
are
objects
that
are
coric/common
to
arbitrary
liftings
of
X
to
Z
p
.
This
observation
is
of
particular
importance
in
light
of
the
strong
structural
resemblances
between
inter-universal
Teichmüller
theory
and
the
theory
of
crystals
[cf.
[Alien],
§3.1,
(v);
the
discussion
of
(CrAND)
in
the
present
§3.5;
the
discussion
of
§3.10
below].
§3.6.
Chains
of
logical
AND
relations
From
the
point
of
view
of
the
simple
qualitative
model
of
inter-universal
Te-
ichmüller
theory
given
in
Example
2.4.5,
the
discussion
of
§3.4
concerns
the
AND
relator
“∧”
in
the
“Θ-link”
portion
of
Example
2.4.5,
(ii).
On
the
other
hand,
strictly
speaking,
this
portion
of
inter-universal
Teichmüller
theory
only
concerns
the
initial
definition
of
the
Θ-link.
That
is
to
say,
the
bulk
of
the
theory
developed
in
[IUTchI-III]
concerns,
from
the
point
of
view
of
the
simple
qualitative
model
of
inter-universal
Teichmüller
theory
given
in
Example
2.4.5,
(ii),
the
preservation
of
the
AND
relator
“∧”
as
one
passes
from
·
the
“Θ-link”
portion
of
Example
2.4.5,
(ii),
to
·
the
“multiradial
representation”
portion
of
Example
2.4.5,
(ii).
By
contrast,
the
passage
from
the
“multiradial
representation”
portion
of
Example
2.4.5,
(ii),
to
the
“final
numerical
estimate”
portion
of
Example
2.4.5,
(ii)
—
i.e.,
which
corresponds
to
the
passage
from
[IUTchIII],
Theorem
3.11,
to
[IUTchIII],
Corollary
3.12
—
is
[cf.
the
discussion
of
the
final
portion
of
Example
2.4.5,
(ii)!]
relatively
straightforward
[cf.
the
discussion
of
§3.10,
§3.11,
below].
At
this
point,
it
is
perhaps
of
interest
to
consider
“typical
symptoms”
of
mathematicians
who
are
operating
under
fundamental
misunderstandings
con-
cerning
the
essential
logical
structure
of
inter-universal
Teichmüller
theory.
Such
typical
symptoms,
which
are
in
fact
closely
related
to
one
another,
include
the
following:
(Syp1)
a
sense
of
unjustified
and
acutely
harsh
abruptness
in
the
pas-
sage
from
[IUTchIII],
Theorem
3.11,
to
[IUTchIII],
Corollary
3.12
[cf.
the
discussion
of
the
final
portions
of
Example
2.4.5,
(ii),
(iii)!];
(Syp2)
a
desire
to
see
the
“proof”
of
some
sort
of
commutative
diagram
or
“compatibility
property”
to
the
effect
that
taking
log-volumes
of
pilot
objects
in
the
domain
and
codomain
of
the
Θ-link
yields
the
same
real
number
[a
property
which,
in
fact,
can
never
be
proved
since
it
is
false!
—
cf.
the
discussion
of
§3.5];
(Syp3)
a
desire
to
see
the
inequality
of
the
final
numerical
estimate
obtained
as
the
result
of
concatenating
some
chain
of
intermediate
inequali-
ties,
i.e.,
as
is
often
done
in
proofs
in
real/complex/functional
analysis
or
analytic
number
theory.
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
105
Here,
it
should
be
noted
that
(Syp2)
and
(Syp3)
often
occur
as
approaches
to
mitigating
the
“harsh
abruptness”
of
(Syp1).
With
regard
to
(Syp3),
it
should
be
emphasized
that
it
is
entirely
unrealistic
to
attempt
to
obtain
the
inequality
of
the
final
numerical
estimate
as
the
result
of
concatenating
some
chain
of
intermediate
inequalities
since
this
is
simply
not
the
way
in
which
the
logical
structure
of
inter-universal
Teichmüller
theory
is
organized.
That
is
to
say,
in
a
word,
the
logical
structure
of
inter-universal
Teichmüller
theory
does
not
proceed
by
concatenating
some
sort
of
chain
of
intermediate
inequalities.
Rather,
(∧-Chn)
the
logical
structure
of
inter-universal
Teichmüller
theory
proceeds
by
observing
a
chain
of
AND
relations
“∧”
[cf.
the
discussion
of
[IUTchIII],
Remark
3.9.5,
(viii),
(ix);
[IUTchIII],
Remark
3.12.2,
(c
itw
),
(f
itw
);
[Alien],
§3.11,
(iv),
(v)].
As
observed
in
Example
2.4.5,
(ii),
(iii),
once
one
follows
this
chain
of
AND
relations
“∧”
up
to
and
including
the
multiradial
representation
of
the
Θ-pilot
[i.e.,
[IUTchIII],
Theorem
3.11],
the
pas-
sage
to
the
final
numerical
estimate
[i.e.,
[IUTchIII],
Corollary
3.12]
is
relatively
straightforward
[i.e.,
as
one
might
expect,
from
the
use
of
the
word
“corollary”!].
One
essentially
formal
consequence
of
(∧-Chn)
is
the
following:
Since
the
defi-
nition
of
the
Θ-link,
the
construction
of
the
multiradial
representation
of
the
Θ-pilot,
and
the
ultimate
passage
to
the
final
numerical
estimate
consist
of
a
finite
number
of
steps,
one
natural
and
effective
way
to
analyze/diagnose
[cf.
the
discussion
of
§1.4!]
the
precise
content
of
misunderstandings
of
inter-universal
Teichmüller
theory
is
to
determine
(∧-Dgns)
precisely
where
in
the
finite
sequence
of
steps
that
appear
is
the
first
step
at
which
the
person
feels
that
the
preservation
of
the
crucial
AND
relator
“∧”
is
no
longer
clear.
In
some
sense,
the
starting
point
of
the
various
AND
relations
“∧”
that
appear
in
the
multiradial
algorithm
of
[IUTchIII],
Theorem
3.11,
is
the
observation
that
(∧-Input)
the
input
data
for
this
multiradial
algorithm
consists
solely
of
an
abstract
F
×μ
-prime-strip;
moreover,
this
multiradial
algorithm
is
functorial
with
respect
to
arbitrary
isomorphisms
between
F
×μ
-prime-
strips
[cf.
[IUTchIII],
Remark
3.11.1,
(ii);
the
final
portion
of
[Alien],
§3.7,
(i)].
This
property
(∧-Input)
means
that
the
multiradial
algorithm
may
be
applied
to
any
F
×μ
-prime-strip
that
appears,
or
alternatively/equivalently,
that
any
F
×μ
-
prime-strip
may
serve
as
the
gluing
data
[cf.
the
“γ
J
”
in
the
analogies
discussed
in
§3.2,
(StR3),
(StR4),
as
well
as
Example
2.4.5,
(ii)!]
between
a
given
situation
[i.e.,
such
as
the
(Θ
±ell
NF-)Hodge
theater
in
the
codomain
of
the
Θ-link!]
and
the
content
of
the
multiradial
algorithm.
On
the
other
hand,
in
order
to
conclude
that
the
multiradial
algorithm
yields
output
data
satisfying
suitable
AND
relations
“∧”,
it
is
necessary
also
to
examine
in
detail
the
content
of
this
output
data,
i.e.,
in
particular,
in
the
context
of
the
central
IPL
and
SHE
properties
discussed
in
[IUTchIII],
Remark
3.11.1,
(iii),
as
well
as
the
chain
of
(sub)quotients
aspect
of
the
SHE
property
[cf.
[IUTchIII],
106
SHINICHI
MOCHIZUKI
Remark
3.11.1,
(iii);
[IUTchIII],
Remark
3.9.5,
(viii),
(ix)].
In
a
word,
the
essential
“principle”
that
is
applied
throughout
the
various
steps
of
the
multiradial
algorithm
in
order
to
derive
new
AND
relations
“∧”
from
old
AND
relations
“∧”
is
the
following
“principle
of
extension
of
indeterminacies”:
(ExtInd)
If
A,
B,
and
C
are
propositions,
then
it
holds
[that
B
=⇒
B
∨
C
and
hence]
that
A
∧
B
=⇒
A
∧
(B
∨
C).
One
important
tool
that
is
frequently
used
in
inter-universal
Teichmüller
theory
in
a
fashion
that
is
closely
related
to
(ExtInd)
is
the
notion
of
a
poly-morphism
[cf.
the
discussion
of
§3.7
below
for
more
details].
In
the
context
of
(ExtInd),
it
is
interesting
to
note
that,
from
the
point
of
view
of
the
discussion
of
§3.4,
the
“∨”
that
appears
in
the
conclusion
—
i.e.,
A
∧
(B
∨
C)
—
of
(ExtInd)
may
be
understood
as
amounting
to
essentially
the
same
phenomenon
as
the
“∨”
that
appears
in
(NeuORInd2)
[e.g.,
by
taking
“C”
to
be
A].
That
is
to
say,
instead
of
generating
AND
relations
“∧”
tautologically
by
means
of
the
introduction
of
distinct
labels
[i.e.,
as
in
(AOΘ1)]
—
i.e.,
say,
by
introducing
a
new
distinct
label
for
“C”
so
as
to
conclude
a
tautological
relation
A
∧
B
∧
C
—
(ExtInd)
allows
one
to
generate
new
AND
relations
“∧”
while
avoiding
the
introduction
of
new
distinct
labels.
As
discussed
in
§3.4,
this
point
of
view
[i.e.,
of
avoiding
the
introduction
of
new
distinct
labels]
leads
inevitably
to
OR
relations
“∨”,
i.e.,
as
in
(NeuORInd2)
or
as
in
the
conclusion
“A
∧
(B
∨
C)”
of
(ExtInd).
As
discussed
above,
the
reason
that
one
wishes
to
avoid
the
introduction
of
new
distinct
labels
when
applying
(ExtInd)
is
precisely
that
(sQLTL)
one
wishes
to
apply
(ExtInd)
to
form
“(sub)quotients/splittings”
of
the
log-theta-lattice
[cf.
the
title
of
[IUTchIII]!],
i.e.,
to
project
the
vertical
line
on
the
left-hand
side
of
the
infinite
“H”
portion
of
the
log-
theta-lattice
onto
the
vertical
line
on
the
right-hand
side
of
this
infinite
“H”
by
somehow
achieving
some
sort
of
“crushing
together”
of
distinct
coordinates
[i.e.,
“(n,
m)”,
where
n,
m
∈
Z]
of
the
log-theta-lattice
[cf.
the
discussion
of
§3.11
below;
[IUTchIII],
Remark
3.9.5,
(viii),
(ix);
[IUTchIII],
Remark
3.12.2,
(c
itw
),
(f
itw
);
[Alien],
§3.11,
(iv),
(v)).
At
this
point,
it
is
of
interest
to
note
that
there
are,
in
some
sense,
two
ways
in
which
(ExtInd)
is
applied
during
the
execution
of
the
various
steps
of
the
multiradial
algorithm
[cf.
the
discussion
of
§3.10,
§3.11,
below,
for
more
details]:
(ExtInd1)
operations
that
consist
of
simply
adding
more
possibilites/indeter-
minacies
[which
corresponds
to
passing
from
B
to
B
∨
C]
within
some
fixed
container;
(ExtInd2)
operations
in
which
one
identifies
[i.e.,
“crushes
together”,
by
passing
from
B
to
B
∨
C]
objects
with
distinct
labels,
at
the
cost
of
passing
to
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
107
a
situation
in
which
the
object
is
regarded
as
being
only
known
up
to
isomorphism.
Typical
examples
of
(ExtInd1)
include
the
upper
semi-continuity
of
(Ind3),
as
well
as
the
passage
to
holomorphic
hulls.
Typically,
such
applications
of
(ExtInd1)
play
an
important
role
in
establishing
various
symmetry
or
invariance
properties
such
as
multiradiality.
This
sort
of
establishment
of
various
symmetry
or
invariance
prop-
erties
by
means
of
(ExtInd1)
then
allows
one
to
apply
label
crushing
operations
as
in
(ExtInd2).
Put
another
way,
·
(ExtInd1)
may
be
understood
as
a
sort
of
operation
whose
purpose
is
to
prepare
suitable
descent
data,
while
·
(ExtInd2)
may
be
thought
of
as
a
sort
of
actual
descent
operation,
i.e.,
from
data
that
depends
on
the
specification
of
a
member
of
some
collection
of
distinct
labels
to
data
that
is
independent
of
such
a
label
specification.
[We
refer
to
the
discussion
of
§3.8
below
for
more
details
on
foundational
aspects
of
(ExtInd2)
and
to
the
discussion
of
§3.9
below
for
more
details
concerning
the
notion
of
“descent”.]
Typical
examples
of
(ExtInd2)
in
inter-universal
Teichmüller
theory
are
the
following
[cf.
the
notational
conventions
of
[IUTchI],
Definition
3.1,
(e),
(f)]:
·
identifying
“Π
v
”’s
[where
v
∈
V]
at
different
vertical
coordinates
[i.e.,
“(n,
m)”
and
“(n,
m
)”,
for
n,
m,
m
∈
Z]
of
the
log-theta-lattice,
which
results
in
a
“Π
v
regarded
up
to
isomorphism”
that
is
labeled
by
a
new
label
“(n,
◦)”;
·
identifying
“G
v
”’s
[where
v
∈
V]
at
different
horizontal
or
vertical
coor-
dinates
[i.e.,
“(n,
m)”
and
“(n
,
m
)”,
for
n,
n
,
m,
m
∈
Z]
of
the
log-theta-
lattice,
which
results
in
a
“G
v
regarded
up
to
isomorphism”
that
is
labeled
by
a
new
label
“(◦,
◦)”;
·
identifying
the
F
×μ
-prime-strips
in
the
Θ-link
that
arise
from
the
Θ-
and
q-pilot
objects
in
distinct
(Θ
±ell
NF-)Hodge
theaters
[i.e.,
the
(Θ
±ell
NF-
)Hodge
theaters
in
the
domain
and
codomain
of
the
Θ-link]
by
working
with
these
F
×μ
-prime-strips
up
to
isomorphism.
In
some
sense,
the
most
nontrivial
instances
of
the
application
of
(ExtInd)
in
the
context
of
the
multiradial
algorithm
occur
in
relation
to
the
log-Kummer-
correspondence
[i.e.,
in
the
vertical
line
on
the
left-hand
side
of
the
infinite
“H”]
and
closely
related
operations
of
Galois
evaluation
[cf.
the
discussion
of
§3.11
below].
The
Kummer
theories
that
appear
in
this
log-Kummer-correspondence
—
i.e.,
Kummer
theories
for
·
multiplicative
monoids
of
nonzero
elements
of
rings
of
integers
in
mixed-
characteristic
local
fields,
·
mono-theta
environments/theta
monoids,
and
·
pseudo-monoids
of
κ-coric
functions
—
involve
the
construction
of
various
[Kummer]
isomorphisms
between
·
Frobenius-like
data
and
·
corresponding
data
constructed
via
anabelian
algorithms
from
étale-like
objects.
108
SHINICHI
MOCHIZUKI
The
output
of
such
algorithms
typically
involves
constructing
the
“corresponding
data”
as
one
possibility
among
many.
Here,
we
note
that
either
of
these
Frobenius-like/étale-like
versions
of
“corresponding
data”
is
—
unlike,
for
instance,
the
data
that
constitutes
an
F
×μ
-prime-strip!
—
sufficiently
robust
that
it
completely
determines
[even
when
only
regarded
up
to
isomorphism!]
the
[usual]
embedding
of
the
Θ-pilot.
That
is
to
say,
taken
as
a
whole,
the
multiradial
algorithm
—
and,
especially,
the
portion
of
the
multiradial
algorithm
that
involves
the
log-Kummer
correspondence
and
closely
related
operations
of
Galois
evaluation
—
plays
the
role
of
exhibiting
the
Frobenius-like
Θ-pilot
as
one
possibility
within
a
collec-
tion
of
possibilities
constructed
via
anabelian
algorithms
from
étale-like
data.
Thus,
in
this
situation,
one
obtains
the
crucial
preservation
of
the
AND
relation
“∧”
by
applying
(ExtInd)
twice,
i.e.,
by
applying
·
(ExtInd1)
to
the
enlargement
of
the
collection
of
possibilities
under
consideration
and
·
(ExtInd2)
to
the
Kummer
isomorphisms
involved,
when
one
passes
from
Frobenius-like
object
labels
“(n,
m)”
[where
n,
m
∈
Z]
to
étale-like
object
labels
“(n,
◦)”
[where
n
∈
Z].
This
is
precisely
what
is
meant
by
the
chain
of
(sub)quotients
aspect
of
the
SHE
property
[cf.
[IUTchIII],
Remark
3.11.1,
(iii);
[IUTchIII],
Remark
3.9.5,
(viii),
(ix)]
discussed
above
[cf.
also
the
discussion
of
§3.10,
§3.11,
below].
§3.7.
Poly-morphisms
and
logical
AND
relations
Poly-morphisms
—
i.e.,
sets
of
morphisms
between
objects
—
appear
through-
out
inter-universal
Teichmüller
theory
as
a
tool
for
facilitating
the
explicit
enumeration
of
a
collection
of
possibilities.
Composable
ordered
pairs
of
poly-morphisms
[i.e.,
pairs
for
which
the
domain
of
the
first
member
in
the
pair
coincides
with
the
codomain
of
the
second
member
in
the
pair]
may
be
composed
by
considering
the
set
of
morphisms
obtained
by
composing
the
morphisms
that
belong
to
the
sets
of
morphisms
that
constitute
the
given
pair
of
poly-morphisms.
Such
compositions
of
poly-morphisms
allow
one
to
keep
track
—
in
a
precise
and
explicit
fashion
—
of
collections
of
possibilities
under
consideration.
From
the
point
of
view
of
chains
of
AND
relations
“∧”,
as
discussed
in
§3.6,
the
collections
of
possibilities
enumerated
by
poly-morphisms
are
to
be
understood
as
being
related
to
one
another
via
OR
relations
“∨”.
That
is
to
say,
poly-morphisms
may
be
thought
of
as
a
sort
of
indeterminacy,
which
is
used
in
inter-universal
Teichmüller
theory
to
produce
structures
that
satisfy
various
symmetry
or
invariance
properties,
hence
yield
suitable
descent
data
[cf.
the
discussion
of
(ExtInd1)
in
§3.6;
the
discussion
of
§3.9
below].
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
109
Thus,
for
instance,
in
the
case
of
the
full
poly-isomorphism
that
constitutes
the
Θ-link,
one
may
understand
the
fundamental
AND
relation
“∧”
of
(AOΘ1)
—
which,
for
simplicity,
we
denote
by
A
∧
B
[where
A
and
B
correspond,
respectively,
in
the
notation
of
the
discussion
of
§3.4,
to
“∗
:→
‡
q-plt
∈
‡
Ring”
and
“∗
:→
†
Θ-plt
∈
†
Ring”]
—
may
be
understood
as
a
relation
“A
∧
(B
1
∨
B
2
∨
.
.
.
)”,
i.e.,
a
relation
to
the
effect
that
if
one
fixes
the
q-pilot
‡
q-plt,
then
this
q-pilot
is
glued,
via
the
Θ-link,
to
the
Θ-pilot
†
Θ-plt
by
means
of
one
isomorphism
[of
the
full
poly-
isomorphism
that
constitutes
the
Θ-link]
or
another
isomorphism,
or
yet
another
isomorphism,
etc.
[Here,
the
various
possible
gluings
that
constitute
B
are
denoted
by
B
1
,
B
2
,
.
.
.
.]
In
particular,
as
discussed
in
(∧-Chn),
if
one
starts
with
the
Θ-link
and
then
con-
siders
various
subsequent
logical
AND
relations
“∧”
that
arise
—
for
instance,
by
considering
various
composites
of
poly-morphisms!
—
by
applying
(ExtInd),
then
(∧(∨)-Chn)
the
essential
logical
structure
of
inter-universal
Teichmüller
theory,
as
discussed
in
(∧-Chn),
may
be
understood
as
follows:
A
∧
B
=
A
∧
(B
1
∨
B
2
∨
.
.
.
)
=⇒
A
∧
(B
1
∨
B
2
∨
.
.
.
∨
B
1
∨
B
2
∨
.
.
.
)
=⇒
A
∧
(B
1
∨
B
2
∨
.
.
.
∨
B
1
∨
B
2
∨
.
.
.
∨
B
1
∨
B
2
∨
.
.
.
)
..
.
Finally,
we
recall
that
various
“classical
examples”
of
the
notion
of
a
poly-
morphism
include
·
the
collection
of
maps
between
topological
spaces
that
constitutes
a
ho-
motopy
class,
or
stable
homotopy
class,
of
maps;
·
the
collection
of
morphisms
between
complexes
that
constitutes
a
mor-
phism
of
the
associated
derived
category;
·
the
collection
of
morphisms
obtained
by
considering
some
sort
of
orbit
by
some
sort
of
group
action
on
the
domain
or
codomain
of
a
given
morphism
[cf.
the
discussion
of
[Rpt2018],
§13,
(PMEx1),
(PMEx2),
(PMQut)].
Also,
in
this
context,
it
is
useful
to
recall
[cf.
the
discussion
of
[Alien],
§4.1,
(iv)]
that
·
gluings
via
poly-morphisms
are
closely
related
to
the
sorts
of
gluings
that
occur
in
the
construction
of
algebraic
stacks
[i.e.,
algebraic
stacks
which
are
not
algebraic
spaces].
§3.8.
Inter-universality
and
logical
AND
relations
One
fundamental
aspect
of
inter-universal
Teichmüller
theory
lies
in
the
con-
sideration
of
distinct
universes
that
arise
naturally
when
one
considers
Galois
110
SHINICHI
MOCHIZUKI
categories
—
i.e.,
étale
fundamental
groups
—
associated
to
various
schemes.
Here,
it
is
important
to
note
that,
when
phrased
in
this
way,
this
fundamental
aspect
of
inter-universal
Teichmüller
theory
is,
at
least
from
the
point
of
view
of
mathematical
foundations,
no
different
from
the
situation
that
arises
in
[SGA1].
On
the
other
hand,
the
fundamental
difference
between
the
situation
considered
in
[SGA1]
and
the
situations
considered
in
inter-universal
Teichmüller
theory
lies
in
the
fact
that,
whereas
in
[SGA1],
the
various
distinct
schemes
that
appear
are
related
to
one
another
by
means
of
morphisms
of
schemes
or
rings,
the
various
distinct
schemes
that
appear
in
inter-universal
Teichmüller
theory
are
related
to
one
another,
in
general,
by
means
of
relations
—
such
as
the
log-
and
Θ-links
—
that
are
non-ring/scheme-theoretic
in
nature,
i.e.,
in
the
sense
that
they
do
not
arise
from
morphisms
of
schemes
or
rings.
In
general,
when
considering
relations
between
distinct
mathematical
objects,
it
is
of
fundamental
importance
to
specify
those
mathematical
structures
that
are
common
—
i.e.,
in
the
terminology
of
inter-universal
Teichmüller
theory,
coric
—
to
the
various
distinct
mathematical
objects
under
consideration.
Here,
we
observe
that
this
notion
of
being
“common”/“coric”
to
the
various
distinct
mathemat-
ical
objects
under
consideration
constitutes,
when
formulated
at
a
formal,
symbolic
level,
a
logical
AND
relation
“∧”.
—
cf.
the
discussion
of
§3.2
[cf.,
especially,
Example
3.2.2],
§3.4,
§3.5,
§3.6,
§3.7.
Thus,
in
the
situations
considered
in
[SGA1],
the
ring/scheme
structures
of
the
various
distinct
schemes
that
appear
are
coric
and
hence
allow
one
to
relate
the
universes/Galois
categories/étale
fundamental
groups
associated
to
these
distinct
schemes
in
a
way
that
makes
use
of
the
common
ring/scheme
structures
between
these
schemes.
At
a
concrete
level,
this
means
that
in
the
situations
considered
in
[SGA1],
étale
fundamental
groups
may
be
related
to
one
another
in
such
a
way
that
the
only
indeterminacies
that
occur
are
inner
automorphism
indeterminacies.
Moreover,
these
inner
automorphism
indeterminacies
are
by
no
means
superfluous
—
cf.
the
discussion
of
Examples
3.8.1,
3.8.2,
3.8.3,
3.8.4
below.
Example
3.8.1:
Inevitability
of
inner
automorphism
indeterminacies.
The
unavoidability
of
inner
automorphism
indeterminacies
may
be
understood
in
very
elementary
terms,
as
follows.
def
(i)
Let
k
be
a
perfect
field;
k
an
algebraic
closure
of
k;
N
⊆
G
k
=
Gal(k/k)
∼
a
normal
closed
subgroup
of
G
k
;
σ
∈
G
k
such
that
the
automorphism
ι
σ
:
N
→
N
of
N
given
by
conjugating
by
σ
is
not
inner.
[One
verifies
immediately
that,
for
instance,
if
k
is
a
number
field
or
a
mixed-characteristic
local
field,
then
such
N
,
σ
do
def
indeed
exist.]
Write
k
N
⊆
k
for
the
subfield
of
N
-invariants
of
k,
G
k
N
=
N
⊆
G
k
,
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
111
def
Q
N
=
G
k
/G
k
N
.
Then
observe
that
this
situation
yields
an
example
of
a
situation
in
which
one
may
verify
directly
that
the
functoriality
of
the
étale
fundamental
group
only
holds
if
one
allows
for
inner
automorphism
indeterminacies
in
the
definition
of
the
étale
fundamental
group.
Indeed,
let
us
first
observe
that
the
“basepoints”
of
k
and
k
N
determined
by
k
allows
us
to
regard
G
k
and
G
k
N
,
respectively,
as
the
étale
fundamental
groups
of
k
and
k
N
.
Thus,
if
one
assumes
that
the
functoriality
of
the
étale
fundamental
group
holds
even
in
the
absence
of
inner
automorphism
indeterminacies,
then
the
commutative
diagram
of
schemes
Spec(k
N
)
σ
−→
Spec(k
N
)
Spec(k)
[where
the
diagonal
morphisms
are
the
natural
morphisms]
induces
a
commutative
diagram
of
profinite
groups
G
k
N
ι
σ
−→
G
k
N
G
k
—
which
[since
the
natural
inclusion
N
=
G
k
N
→
G
k
is
injective!]
implies
that
ι
σ
is
the
identity
automorphism,
in
contradiction
to
our
assumption
concerning
σ!
(ii)
The
phenomenon
discussed
in
(i)
may
be
understood
as
a
consequence
of
the
fact
that,
whereas
Spec(k)
is
coric
in
the
commutative
diagram
of
schemes
that
appears
in
(i)
[i.e.,
in
the
sense
that
this
diagram
does
indeed
commute!],
Spec(k)
is
not
coric
in
the
diagram
of
schemes
Spec(k)
Spec(k
N
)
σ
−→
Spec(k
N
)
Spec(k)
[where
the
diagonal
morphisms
are
the
natural
morphisms],
i.e.,
in
the
sense
that
the
upper
portion
of
this
diagram
does
not
commute!
(iii)
Finally,
we
consider
the
natural
exact
sequence
1
−→
G
k
N
−→
G
k
−→
Q
N
−→
1
of
profinite
groups.
Then
observe
that
the
inner
automorphisms
indeterminacies
of
G
k
[cf.
the
discussion
of
(i),
(ii)!]
induce
outer
automorphism
indeterminacies
of
G
k
N
that
will
not,
in
general,
be
inner.
That
is
to
say,
112
SHINICHI
MOCHIZUKI
if
one
considers
G
k
N
in
the
context
of
this
natural
exact
sequence,
then
one
must
in
fact
consider
G
k
N
[not
only
up
to
inner
automorphism
in-
determinacies,
i.e.,
as
discussed
in
(i),
(ii),
but
also]
up
to
certain
outer
automorphism
indeterminacies.
Relative
to
the
point
of
view
of
the
discussion
of
(ii),
these
outer
automorphism
indeterminacies
may
be
understood
as
a
consequence
of
the
fact
that,
in
the
context
of
the
field
extensions
k
⊆
k
N
⊆
k
and
the
automorphisms
of
these
field
extensions
induced
by
elements
of
G
k
,
the
field
k
is
coric,
whereas
the
field
k
N
is
not
coric
—
i.e.,
in
the
context
of
these
field
extensions
and
automorphisms
of
field
extensions,
the
relationship
of
k
to
the
various
field
extensions
that
appear
is
constant
and
fixed,
whereas
the
relationship
of
k
N
to
the
various
field
extensions
that
appear
is
variable,
i.e.,
subject
to
indeterminacies
arising
from
the
action
of
elements
of
G
k
.
Example
3.8.2:
Inter-universality
and
the
structure
of
(Θ
±ell
NF-)Hodge
theaters.
In
the
following
discussion
of
(Θ
±ell
NF-)Hodge
theaters,
we
fix
a
col-
lection
of
initial
Θ-data
(F
/F,
X
F
,
l,
C
K
,
V,
V
bad
mod
,
)
as
in
[IUTchI],
Definition
3.1,
and
apply
the
notational
conventions
of
[IUTchI],
def
Definition
3.1.
In
particular,
we
recall
that
E
=
E
F
is
an
elliptic
curve
over
the
number
field
F
;
F
is
an
algebraic
closure
of
F
;
l
is
a
prime
number;
K
⊆
F
is
the
extension
field
of
F
determined
by
the
composite
of
the
fields
of
definition
of
the
closed
points
of
the
finite
group
scheme
E[l]
⊆
E
of
l-torsion
points
of
E;
F
mod
⊆
F
is
the
field
of
moduli
of
E,
i.e.,
the
field
extension
of
the
field
of
rational
numbers
generated
by
the
j-invariant
of
E.
For
simplicity,
we
assume
that
l
>
5.
(i)
We
begin
by
recalling
the
following:
(i-a)
The
point
of
view
of
classical
Galois
theory
with
regard
to
constructing
finite
Galois
extensions
of
fields
may
be
summarized,
in
the
case
of
the
Galois
extension
K/F
,
as
follows:
·
one
starts
with
a
base
field
F
;
·
one
then
constructs
a
finite
field
extension
K
of
F
that
is
saturated
with
respect
to
Galois
conjugation
over
F
.
Thus,
relative
to
this
classical
point
of
view,
one
is
constrained
to
viewing
the
situation
from
the
point
of
view
of
the
base
field
F
.
This
constraint
obliges
one
to
always
take
into
account
the
entirety
of
Galois
conjugates
[over
F
]
of
objects
associated
to
K.
The
point
of
view
of
(i-a)
is
fundamentally
incompatible
with
the
main
goal
of
the
construction
of
(Θ
±ell
NF-)Hodge
theaters
in
[IUTchI],
namely,
the
simulation
of
a
global
multiplicative
subspace
of
E[l]
[cf.
the
discussion
of
global
multi-
plicative
subspaces
in
[IUTchI],
§I1;
[Alien],
§2.3,
§2.4;
[Alien],
§3.3,
(iv),
as
well
as
Example
3.2.1,
(vi),
of
the
present
paper],
together
with
a
global
canonical
generator,
up
to
±1,
of
the
quotient
of
E[l]
by
the
global
multiplicative
subspace
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
113
[cf.
the
discussion
of
global
canonical
generators
in
[IUTchI],
§I1;
[Alien],
§3.3,
(iv),
as
well
as
Example
3.2.1,
(vi),
of
the
present
paper].
In
some
sense,
the
technical
starting
point
of
the
“simulation
of
a
global
multiplicative
subspace”
implemented
in
[IUTchI]
may
be
summarized
as
follows:
(i-b)
The
“simulation
of
a
global
multiplicative
subspace”
given
in
[IUTchI]
is
achieved
by,
in
some
sense,
reversing
the
flow
of
the
classical
construction
reviewed
in
(i-a),
i.e.,
by
·
viewing
the
situation
[not
from
the
point
of
view
of
the
base
field
F
,
but
rather]
from
the
point
of
view
of
the
hyperbolic
orbicurve
C
K
—
which
may
be
thought
of
as
data
that
amounts
to
K,
together
with
a
fixed
choice
of
a
quotient
“Q”
[cf.
[IUTchI],
Definition
3.1,
(f)]
of
E[l],
i.e.,
whose
kernel
is
to
serve
as
the
“simulated
global
multiplicative
subspace”
—
and
·
regarding
the
base
field
F
mod
—
or,
at
the
level
of
hyperbolic
orbicurves,
C
F
mod
[cf.
[IUTchI],
Remark
3.1.7,
(i)]
—
as
a
finite
étale
quotient
of
K
[or,
at
the
level
of
hyperbolic
orbicurves,
C
K
],
i.e.,
which
amounts
to
thinking
in
terms
of
[compatible]
finite
étale
quotients
Spec(K)
Spec(F
mod
),
C
K
→
C
F
mod
—
which
are
regarded
as
objects
constructed
from
Spec(K),
C
K
.
This
approach
allows
one
to
concentrate
on
a
fixed
[simulated
global
mul-
tiplicative]
subspace
and
hence
[unlike
the
situation
discussed
in
(i-a)!]
to
exclude
the
various
nontrivial
Galois
conjugates
over
F
of
this
fixed
simulated
global
multiplicative
subspace.
The
approach
of
(i-b)
has
numerous
important
technical
consequences
[to
be
dis-
cussed
in
(ii),
(iii),
(iv),
below].
(ii)
From
the
point
of
view
of
étale-like
objects
[i.e.,
arithmetic
fundamental
groups],
constructing
a
quotient
C
K
→
C
F
mod
as
in
(i-b)
corresponds
to
constructing
a
profinite
group
“Π
C
F
mod
”
from
the
profinite
group
Π
C
K
that
contains
Π
C
K
as
an
open
subgroup.
In
light
of
the
well-known
slimness
of
Π
C
F
mod
[cf.,
e.g.,
[AbsTopI],
Proposition
2.3,
(ii)],
such
a
construction
of
“Π
C
F
mod
”
amounts
to
the
construction
of
a
finite
group
of
outer
automorphisms
of
some
open
subgroup
of
Π
C
K
.
This
finite
group
may
be
thought
of
as
a
finite
quotient
group
Π
C
F
mod
Γ
mod
of
Π
C
F
mod
.
If
we
think
of
the
absolute
Galois
group
G
F
mod
of
the
number
field
F
mod
as
a
quotient
Π
C
F
mod
G
F
mod
of
Π
C
F
mod
,
then
this
finite
quotient
group
Γ
mod
determines
a
finite
quotient
group
G
F
mod
Γ
Gal
mod
of
G
F
mod
.
Here,
we
recall
from
the
construction
of
[IUTchI],
Example
4.3,
(i),
that
Γ
Gal
mod
has
a
natural
subquotient
that
may
be
×
identified
with
F
l
=
F
l
/{±1},
i.e.,
which
corresponds
to
the
multiplicative
F
l
-
symmetry
of
the
(Θ
±ell
NF-)Hodge
theater.
In
particular,
Γ
Gal
mod
,
hence
also
Γ
mod
,
is
a
finite
group
of
order
>
2,
which
implies,
by
well-known
properties
of
absolute
Galois
groups
of
number
fields
[cf.,
e.g.,
[NSW],
Theorem
12.1.7]
that
def
114
SHINICHI
MOCHIZUKI
(NoSpl)
The
surjection
G
F
mod
Γ
Gal
mod
of
profinite
groups
does
not
admit
a
splitting.
Here,
we
note
that
[in
light
of
the
well-known
slimness
of
G
F
mod
—
cf.,
e.g.,
[Ab-
sTopI],
Theorem
1.7,
(iii)]
this
non-existence
of
a
splitting
may
be
reformulated
as
the
assertion
that
the
natural
outer
action
of
Γ
Gal
mod
on
the
kernel
Ker(G
F
mod
Gal
Gal
Γ
mod
)
does
not
admit
a
lifting
to
an
action
of
Γ
mod
on
Ker(G
F
mod
Γ
Gal
mod
),
i.e.,
to
an
action
that
is
free
of
inner
automorphism
indeterminacies.
In
particular,
it
follows
[a
fortiori!]
that
the
natural
outer
action
of
Γ
mod
on
Ker(Π
C
F
mod
Γ
mod
)
does
not
admit
a
lifting
to
an
action
of
Γ
mod
on
Ker(Π
C
F
mod
Γ
mod
),
i.e.,
to
an
action
that
is
free
of
inner
automorphism
indeterminacies.
That
is
to
say,
in
sum-
mary,
the
inner
automorphism
indeterminacies
in
these
natural
outer
actions
are
essential
and
unavoidable.
(iii)
The
existence
of
the
inner
automorphism
indeterminacies
discussed
in
(ii)
implies,
in
particular,
that
the
permutations
of
prime-strips
in
the
multiplicative
symmetry
portion
of
a
(Θ
±ell
NF-)Hodge
theater
induced
by
the
F
l
-symmetries
±ell
of
the
(Θ
NF-)Hodge
theater
necessarily
give
rise
to
inner
automorphism
indeter-
minacies
in
the
isomorphisms
between
the
copies
of
local
absolute
Galois
groups
G
v
[where
v
∈
V
non
]
that
appear
in
prime-strips
with
distinct
labels
∈
F
l
[cf.
[IUTchI],
Remark
4.5.1,
(iii);
[IUTchII],
Remark
2.5.2,
(iii);
[IUTchII],
Remarks
4.7.2,
4.7.6;
[Alien],
§3.6,
(iii)].
Put
another
way,
(NoSyn)
there
is
no
well-defined
synchronization
between
these
copies
of
G
v
that
appear
in
prime-strips
at
distinct
labels
∈
F
l
that
is
free
of
inner
automorphism
—
i.e.,
conjugacy
—
indeterminacies.
In
this
context,
we
recall
that
such
a
conjugate
synchronization
is
of
fundamen-
tal
importance
in
inter-universal
Teichmüller
theory
since
it
is
necessary
in
order
to
construct
the
data
that
appears
in
the
unit
group
portion
of
the
F
×μ
-prime-strip
that
appears
in
the
domain
of
the
Θ-link,
i.e.,
data
that
is
required
to
be
free
of
any
dependence
on
the
distinct
labels
∈
F
l
.
Such
a
conjugate
synchronization
is
±
=
F
l
{±1},
achieved
by
applying
the
F
±
l
-symmetries
[where
we
recall
that
F
l
i.e.,
relative
to
the
natural
action
of
{±1}
on
F
l
]
in
the
additive
symmetry
portion
of
the
(Θ
±ell
NF-)Hodge
theater
under
consideration
[cf.
[IUTchII],
Corollary
3.5,
(i);
[IUTchII],
Remark
3.5.2,
(iii);
[IUTchII],
Remark
4.5.3,
(i);
[IUTchIII],
Theorem
1.5,
(iii);
[IUTchIII],
Remark
1.5.1,
(i);
[Alien],
§3.6,
(ii)].
Here,
we
observe
that
in
order
to
achieve
this
conjugate
synchronization
via
the
F
±
l
-symmetry
of
the
various
copies
of
G
v
that
appear
in
prime-strips
with
distinct
labels,
it
is
of
fundamental
importance
to
keep
these
copies
of
G
v
isolated
from
the
absolute
Galois
groups
of
number
fields
that
appear
in
the
discussion
of
(ii)
[i.e.,
since,
as
observed
in
(ii),
it
is
precisely
the
intrinsic
structure
of
these
global
absolute
Galois
groups
that
gives
rise
to
the
unwanted
inner
automorphism/conjugacy
indeterminacies!].
This
local-global
isolation
requirement
—
i.e.,
in
effect,
the
requirement
that
def
(LGIsl)
these
copies
of
the
local
absolute
Galois
group
G
v
be
regarded
not
as
subgroups
of
some
global
absolute
Galois
group,
but
rather
as
coric
objects
that
are
treated
as
being
independent
of
any
sort
of
embedding
into
a
global
absolute
Galois
group
[cf.
[IUTchII],
Remark
4.7.6;
[Alien],
§3.6,
(iii)]
—
will
have
important
consequences,
as
we
shall
see
in
the
discussion
of
(iv)
below.
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
115
(iv)
As
discussed
in
(iii),
the
issue
(SymIsl)
of
isolating
the
F
±
l
-symmetry
from
the
F
l
-symmetry
in
order
to
achieve
conjugate
synchronization
is
one
important
reason
for
imposing
the
local-global
isolation
requirement
(LGIsl)
in
the
context
of
the
construction
of
(Θ
±ell
NF-)Hodge
theaters.
In
fact,
however,
this
property
(LGIsl)
is
fundamental
to
the
entire
structure
of
a
(Θ
±ell
NF-)Hodge
theater
[cf.,
[IUTchI],
Fig.
6.5;
[Alien],
Fig.
3.8].
That
is
to
say,
the
issue
(SymIsl)
may
be
thought
of
as
being
reflected
in
the
gluing
along
certain
collections
of
prime-
strips
between
the
additive
and
multiplicative
symmetry
portions
of
the
(Θ
±ell
NF-
)Hodge
theater
[cf.,
[IUTchI],
Fig.
6.5;
[Alien],
Fig.
3.8].
In
fact,
however,
(SctNF)
even
within
the
multiplicative
symmetry
portion
of
a
(Θ
±ell
NF-)Hodge
theater,
the
goal
of
simulating
a
global
canonical
generator
requires
one
to
treat
the
various
prime-strips
that
appear
in
the
multiplicative
symmetry
portion
of
the
(Θ
±ell
NF-)Hodge
theater
as
“sections”,
in
some
suitable
sense,
of
the
finite
étale
quotient
Spec(K)
Spec(F
mod
)
—
a
point
of
view
that
is
fundamentally
incompatible
with
the
prime
decompo-
sition
trees
of
the
number
fields
K,
F
mod
,
hence
again
requires
one
to
impose
(LGIsl)
[cf.
[IUTchI],
Remarks
4.3.1,
4.3.2;
[Alien],
§3.3,
(iv)].
On
the
other
hand,
let
us
recall
that
the
ring
structure
of
the
nonarchimedean
local
field
that
gives
rise
to
G
v
cannot
be
reconstructed
from
the
abstract
topological
group
G
v
[cf.
[NSW],
the
Closing
Remark
preceding
Theorem
12.2.7;
[AbsTopIII],
§I3;
[Alien],
Example
2.12.3,
(i)].
In
particular,
once
one
imposes
(LGIsl),
the
crucial
reconstruction
of
the
ring
structures
of
the
nonarchimedean
local
fields
that
give
rise
to
the
various
copies
of
G
v
—
where
we
recall
that
such
ring
structures
play
a
fundamental
and
indispensable
role
in
the
definition
of
the
log-link!
—
can
only
be
conducted
if
one
applies
the
absolute
anabelian
algorithms
of
[AbsTopIII],
§1,
locally
at
each
v
∈
V
non
[not
to
G
v
,
but
rather]
to
Π
v
,
i.e.,
one
must
always
regard
each
coric
copy
of
G
v
as
a
“certain
quotient”
of
a
corresponding
coric
copy
of
Π
v
.
Indeed,
from
a
historical
point
of
view
[cf.
the
discussion
of
[IUTchI],
Remark
4.3.2],
it
was
precisely
these
local-global
isolation
aspects
—
i.e.,
surround-
ing
(LGIsl),
as
discussed
in
(iii)
and
the
present
(iv)
—
of
the
structure
of
(Θ
±ell
NF-)Hodge
theaters
that
motivated
the
author
to
develop
the
absolute
anabelian
algorithms
of
[AbsTopIII],
§1,
in
the
first
place!
Example
3.8.3:
Truncated
vs.
profinite
Kummer
theory
and
compatibil-
ity
with
the
p-adic
logarithm.
In
the
following
discussion,
we
fix
notation
as
follows:
Let
k
be
a
finite
extension
of
Q
p
,
for
some
prime
number
p;
k
an
algebraic
closure
of
k.
Write
def
·
G
k
=
Gal(k/k);
·
O
k
for
the
ring
of
integers
of
k,
with
maximal
ideal
m
k
⊆
O
k
;
·
O
⊆
O
k
for
the
multiplicative
monoid
of
nonzero
elements
of
O
k
;
k
·
O
×
⊆
O
for
the
group
of
invertible
elements
of
O
;
k
k
k
116
SHINICHI
MOCHIZUKI
·
O
×
O
×μ
for
the
quotient
of
O
×
by
the
subgroup
μ
∞
of
torsion
k
k
k
elements
[i.e.,
roots
of
unity]
of
O
×
;
k
×
·
log
k
:
O
k
for
the
p-adic
logarithm
on
O
×
.
k
k
×μ
×
O
⊆
O
⊆
O
k
.
Let
Π
G
k
be
a
topological
Thus,
G
k
acts
naturally
on
O
k
k
k
group
equipped
with
a
surjection
onto
G
k
,
which
determines
natural
actions
of
Π
on
O
×μ
O
×
⊆
O
⊆
O
k
.
We
shall
often
think
of
the
pair
G
k
O
or
the
pair
k
k
k
k
Π
O
“abstractly”
as
a
pair
consisting
of
an
abstract
topological
monoid
[i.e.,
k
O
]
equipped
with
a
continuous
action
by
an
abstract
topological
group
[i.e.,
G
k
k
or
Π].
Also,
we
shall
write
N
≥1
for
the
multiplicative
monoid
of
positive
natural
numbers.
(i)
The
three
types
of
Kummer
theory
that
occur
in
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
Example
3.8.4,
(i-a),
(i-b),
(i-c),
below]
involve
Kum-
mer
towers
that
consist
of
N
-th
power
maps,
for
N
∈
N
≥1
,
on
the
monoids
involved.
In
the
case
of
the
pair
G
k
O
,
one
observes
immediately
that
such
k
N
-th
power
maps
satisfy
the
following
properties
[cf.
the
discussion
of
[IUTchII],
Remark
3.6.4,
(i)],
where
we
assume
that
N
≥
2:
(i-a)
the
N
-th
power
map
O
O
is
not
a
ring
homomorphism,
i.e.,
is
not
k
k
compatible
with
the
additive
structure
underlying
the
ring
structure
on
O
∪
{0};
k
(i-b)
the
N
-th
power
map
O
O
is
(Π
)
G
k
-equivariant,
hence
may
be
k
k
thought
of
as
inducing
on
étale-like
cyclotomes
constructed
via
anabelian
algorithms
from
G
k
or
Π
the
isomorphism
functorially
induced
by
some
—
at
least
from
an
a
priori
point
of
view
—
indeterminate
automor-
phism
[cf.
(i-a),
which
implies
that
G
k
or
Π
must
be
treated
as
abstract
topological
groups,
that
is
to
say,
as
opposed
to
groups
of
ring/field
auto-
morphisms,
i.e.,
“Galois
groups/arithmetic
fundamental
groups”;
the
dis-
cussion
preceding
Example
3.8.1;
the
discussion
following
Example
3.8.4
below;
the
discussion
of
(vi-c)
below]
of
G
k
or
Π;
(i-c)
the
N
-th
power
map
O
O
alters,
at
least
from
an
a
priori
point
of
k
k
view,
cyclotomic
rigidity
isomorphisms
—
but
not
synchronizations
between
collections
of
cyclotomic
rigidity
isomorphisms!
—
between
étale-
like
cyclotomes
[cf.
(i-b)]
and
[Frobenius-like]
cyclotomes
arising
from
the
torsion
subgroup
of
O
—
cf.,
e.g.,
the
[in
general]
nontrivial
action
of
k
the
N
-th
power
map
on
the
n-th
roots
of
unity
in
O
for
n
∈
N
≥1
prime
k
to
N
.
Note
that
it
follows
from
(i-a)
that,
if
we
think
of
a
“basepoint”
as
a
particular
“rigid”
choice
of
an
algebraic
closure
that
is
free
of
any
conjugacy
or
N
-th
power
map
indeterminacies,
then
whereas
(i-d)
the
p-adic
logarithm
k
⊇
O
⊇
O
×
k
k
log
k
−→
k
⊇
O
⊇
O
×
k
k
yields
a
precise,
well-defined
—
and,
in
particular,
free
of
any
conjugacy
or
N
-th
power
map
indeterminacies!
—
set-theoretic
[but
not
ring-theoretic!]
relationship
between
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
117
·
the
basepoint
of
the
[“abstract”]
copy
of
G
k
O
in
the
domain
k
of
log
k
and
·
the
basepoint
of
the
[“abstract”]
copy
of
G
k
O
in
the
k
codomain
of
log
k
[where
we
think
of
both
of
these
copies
of
G
k
O
as
constituent
objects
k
in
the
respective
Kummer
towers
in
the
domain/codomain
of
log
k
,
as
discussed
above],
i.e.,
a
single
unified
basepoint
that
is
simultaneously
valid
for
both
the
domain
and
codomain
of
log
k
,
(i-e)
the
map
Log
k
induced
by
the
p-adic
logarithm
log
k
on
inverse
limits
of
the
Kummer
tower
×
lim
←−
O
k
⊇
lim
←−
O
k
←−
k
⊇
lim
N
N
N
Log
k
−→
k
⊇
O
⊇
O
×
k
k
[where
the
inverse
limits
are
over
N
∈
N
≥1
,
and
we
recall
that
raising
to
the
N
-th
power
on
the
“O
”
in
the
domain
of
log
k
corresponds
to
multiplying
k
by
N
on
the
“k”
in
the
codomain
of
log
k
]
only
yields
a
relationship
between
·
the
inverse
limit
basepoint
in
the
domain
of
Log
k
and
·
the
basepoint
associated
to
a
single
constituent
Kummer
tower
object
[with
a
fixed
additive
structure!]
in
the
codomain
of
Log
k
×
[i.e.,
“O
×
”
as
opposed
to
“lim
←−
O
”].
k
N
k
Note,
moreover,
that
it
follows
from
(i-b),
(i-c)
that
the
basepoint
shifts
that
occur
as
one
passes
between
different
constituent
Kummer
tower
objects
via
various
N
-th
power
maps
are
indeed
—
at
least
from
an
a
priori
point
of
view
—
substan-
tive/nontrivial
in
the
context
of
cyclotomic
rigidity
isomorphisms
or
synchroniza-
tions
between
cyclotomes.
(ii)
The
situation
discussed
in
(i-e)
may
be
understood
as
a
consequence
of
the
fact
that
(ii-a)
the
Kummer
tower
inverse
limit
“lim
←−
”
[where
we
omit
the
subscript
N
to
simplify
notation]
is
biased
toward
the
multiplicative
structure
of
the
rings
involved
—
i.e.,
at
the
expense
of
the
additive
structures
of
these
rings
[cf.
(i-a)]
—
hence
fundamentally
incompatible
with
the
“juggling/rotation/permutation”
of
the
additive
and
multiplicative
structures
that
arises
from
the
log-link
[cf.
the
discussion
of
Example
3.3.2,
(iv)].
In
the
situation
of
(i-e),
we
observe,
moreover,
that
the
problem
of
constructing
×
some
sort
of
“hyper-multiplicative
tower”
of
copies
of,
say,
the
←
lim
−
O
k
in
the
domain
of
Log
k
that
lifts
[i.e.,
relative
to
Log
k
]
the
[multiplicative]
Kummer
tower
of
copies
of
O
×
⊆
O
⊆
k
in
the
codomain
of
Log
k
appears
to
be
unrealistically
intractable:
k
k
Indeed,
compatibility,
relative
to
Log
k
,
with
the
[multiplicative]
Kummer
tower
118
SHINICHI
MOCHIZUKI
in
the
codomain
of
Log
k
would
imply
that
the
transition
maps
of
such
a
“hyper-
multiplicative
tower”
would,
at
least
at
a
purely
formal
computational
level
(!),
necessarily
be
of
the
form
x
=
exp(log(x))
→
exp({log(x)}
M
)
=
exp({log(x)}
·
{log(x)}
M
−1
)
=
x
{log(x)}
M
−1
—
where
M
∈
N
≥1
,
and
the
notation
“exp(−)”
and
“log(−)”
is
intended
in
a
purely
formal
computational
sense
(!).
On
the
other
hand,
(ii-b)
it
seems
difficult
to
conceive
of
any
sort
of
natural
approach
to
construct-
ing
such
“hyper-multiplicative
transition
maps”
×
×
lim
←−
O
k
←−
O
k
→
lim
M
−1
that
realize
the
purely
formal
computation
“x
→
x
{log(x)}
”
for
×
and,
moreover,
allow
one
to
relate,
in
some
natural
way,
the
x
∈
lim
O
←−
k
×
[multiplicative]
Kummer
theory
associated
to
the
←
lim
−
O
k
in
the
do-
main
of
the
transition
map
to
the
corresponding
[multiplicative]
Kummer
×
theory
associated
to
the
←
lim
−
O
in
the
codomain
of
the
transition
map.
k
(iii)
Note
that
the
conditions
imposed
on
the
“hyper-multiplicative
transition
maps”
in
(ii-b)
are
stated
in
a
somewhat
rough
and
imprecise
way.
Although
it
is
not
clear
at
the
time
of
writing
how
to
make
these
conditions
completely
precise,
it
does,
however,
seem
natural
to
consider
the
possibility
of
the
existence
of
commutative
diagrams
as
in
(iii-a)
below,
i.e.,
where
one
thinks
of
“Z”
as
a
sort
of
candidate
for
the
inverse
limit
of
the
“hyper-multiplicative
transition
maps”
of
(ii-b).
That
is
to
say,
the
existence
of
such
a
commutative
diagram
may
be
thought
of
as
a
sort
of
necessary
condition
for
the
existence
of
a
suitable
system
of
“hyper-multiplicative
transition
maps”
as
in
(ii-b).
[Here,
we
note
that
the
surjectivity
condition
of
(iii-
a)
below
may
be
understood
as
a
sort
of
“very
weak
necessary”
version
of
the
domain/codomain
Kummer
theory-relatability
condition
of
(ii-b).]
In
fact,
however,
we
observe
that,
in
the
situation
of
(ii-b)
[and
the
surrounding
discussion],
such
a
commutative
diagram
does
not
exist:
×
(iii-a)
Let
Z
be
a
set
with
an
action
by
G
k
,
ζ
:
Z
→
lim
←−
O
k
a
G
k
-equivariant
map
of
sets
that
induces
a
surjection
from
the
J-invariants
in
the
domain
of
ζ
to
the
J-invariants
in
the
codomain
of
ζ
for
every
closed
subgroup
J
⊆
G
k
that
acts
trivially
on
μ
∞
⊆
k.
[Thus,
ζ
itself
is
necessarily
surjective.]
Then
there
does
not
exist
any
commutative
diagram
of
the
form
λ
−→
lim
k
Z
←−
⏐
⏐
⏐
ζ
⏐
ψ
Log
k
×
lim
k
←−
O
k
−→
—
where
λ
is
a
G
k
-equivariant
map
of
sets,
and
ψ
is
the
natural
projection.
(iii-b)
The
map
Log
k
does
not
admit
any
factorization
×
lim
←−
O
k
N
λ
−→
lim
←−
k
N
ψ
−→
k
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
119
—
where
λ
is
a
G
k
-equivariant
map
of
sets,
and
ψ
is
the
natural
projection.
Indeed,
since
(iii-b)
follows
formally
from
(iii-a),
it
suffices
to
verify
(iii-a).
Suppose
that
a
commutative
diagram
as
in
(iii-a)
exists.
Write
·
O
k
=
O
k
∩
k,
O
k
×
=
O
×
∩
k,
m
k
=
m
k
∩
k;
def
def
def
k
def
·
F
=
k(μ
∞
)
⊆
k.
Let
x
∈
p
2
·
O
k
be
a
nonzero
element.
Write
·
f
=
1
+
x
∈
O
k
×
,
(0
=)
y
=
log
k
(f
)
∈
m
k
;
·
E
⊆
k
for
the
field
extension
of
F
obtained
by
adjoining
all
N
-th
roots
of
f
,
for
N
∈
N
≥1
,
in
k.
def
def
Thus,
f
lifts
[via
the
natural
projection]
to
an
element
of
the
codomain
of
ζ
that
def
is
fixed
by
the
action
of
J
=
Gal(k/E)
⊆
G
k
,
hence
also
[by
our
surjectivity
assumption
on
ζ]
to
an
element
f
Z
∈
Z
of
the
domain
of
ζ
that
is
fixed
by
the
action
of
J.
Since
λ
is
G
k
-equivariant,
we
thus
conclude
that
λ(f
Z
)
is
fixed
by
the
action
of
J
and
maps
via
the
natural
projection
ψ
to
an
element
z
∈
E
⊆
k
that
defines
a
divisible,
hence
l-divisible
element
of
E
×
,
for
any
prime
number
l
=
p.
On
the
other
hand,
it
follows
immediately
from
the
commutativity
of
the
diagram
that
y
=
z.
Next,
observe
that
since
[the
unit!]
f
is
already
clearly
l-divisible
in
k
×
,
hence
also
in
F
×
,
the
Galois
group
Gal(E/F
)
is
isomorphic
to
a
quotient
of
Z
p
.
But
this
implies
that
all
l-power
roots
of
[the
non-unit!]
y
=
z
∈
k
×
⊆
F
×
are
contained
in
F
,
in
contradiction
to
the
easily
verified
fact
that
the
value
group
of
the
valued
field
F
is
isomorphic
to
Z[p
−1
].
This
completes
the
proof
of
(iii-a).
(iv)
The
qualitatively
different
behavior
that
occurs
in
(i-d)
and
(i-e)
may
be
understood
as
being
a
consequence
of
the
fact
[cf.
(i-a)]
that
whereas
(iv-a)
the
ring
structure
on
O
∪
{0}
[which
makes
it
possible
to
define
k
the
well-known
power
series
for
the
p-adic
logarithm
log
k
]
remains
intact
at
any
particular
constituent
Kummer
tower
object
[cf.
the
situation
of
(i-d)],
(iv-b)
the
natural
multiplicative
structure
on
×
lim
←−
O
k
⊇
lim
←−
O
k
←−
k
⊇
lim
N
N
N
does
not
admit
any
corresponding
additive
structure
that
gives
rise
to
a
ring
structure
[i.e.,
that
would
make
it
possible
to
define
the
well-
known
power
series
for
the
logarithm,
hence
a
factorization
as
in
(iii-b)]
on
any
of
the
three
inverse
limits
in
the
above
display
that
is
compatible
with
the
natural
action
by
G
k
and
the
various
natural
projections
to
k
[cf.
the
situation
of
(i-e)].
Moreover,
we
observe
that
(iv-c)
the
various
natural
G
k
-equivariant
projections
of
multiplicative
monoids
lim
←−
k
k;
N
lim
←−
O
k
O
k
;
N
×
×
lim
←−
O
k
O
k
N
120
SHINICHI
MOCHIZUKI
do
not
admit
splittings
[as
may
be
seen,
for
instance,
by
restricting
such
a
splitting
to
the
roots
of
unity,
where
the
existence
of
such
a
splitting
would
amount,
in
particular,
to
a
splitting
of
the
natural
surjection
Q
p
Q
p
/Z
p
].
That
is
to
say,
there
is
no
natural
way
to
relate
the
finite
Kummer
theory
for
a
single
constituent
Kummer
tower
object
“k”,
“O
”,
“O
×
”
in
the
codomain
of
the
map
k
k
Log
k
of
(i-e)
to
the
corresponding
profinite
Kummer
theory
obtained
by
passing
to
the
inverse
limit
“lim
←−
”
of
the
associated
Kummer
tower.
N
(v)
In
the
context
of
(iv-b)
[and
the
surrounding
discussion],
it
is
also
of
interest
to
observe
that
in
fact
(v-a)
neither
of
the
inverse
limits
def
def
lim
I
k
=
←
−
k;
I
O
×
=
lim
←−
k
N
N
O
×
k
∪
{0}
admits
a
field
structure
that
is
stabilized
by
the
natural
action
of
G
k
,
and
whose
underlying
multiplicative
structure
is
the
natural
multiplicative
structure
on
the
inverse
limit.
Indeed,
suppose
that
I
∈
{I
k
,
I
O
×
}
admits
such
a
field
structure.
Let
l
be
an
odd
k
prime
that
does
not
divide
the
order
of
the
finite
group
of
roots
of
unity
of
k.
Write
μ
l
⊆
k
for
the
group
of
l-th
roots
of
unity
of
k,
(μ
l
⊆)
μ
l
∞
⊆
k
for
the
group
of
def
def
def
l-power
roots
of
unity
of
k,
K
=
k(μ
l
∞
)
⊆
k,
G
k
⊇
G
K
=
Gal(k/K),
G
K/k
=
def
G
k
/G
K
,
L
=
I
G
K
[i.e.,
the
subfield
of
G
K
-invariants
of
I].
Then
our
assumption
on
l
implies
that
the
natural
faithful
action
of
G
K/k
on
μ
l
∞
[which
allows
us
to
think
of
G
K/k
as
a
closed
subgroup
of
Z
×
l
]
induces
a
nontrivial
action
of
G
K/k
on
×
∼
μ
l
,
hence
[in
light
of
the
well-known
structure
of
the
profinite
group
Z
×
l
=
Z
l
×
F
l
for
odd
primes
l]
that
G
K/k
contains
a
nontrivial
finite
closed
subgroup
H
⊆
G
K/k
.
Next,
observe
that
the
group
of
divisible
elements
of
the
multiplicative
module
K
×
is
equal
to
μ
l
∞
[cf.
the
fact
that
G
K/k
is
isomorphic
to
a
closed
subgroup
of
Z
×
l
;
[Tsjm],
Lemma
D,
(iii),
(iv)].
This
implies
that
the
multiplicative
G
K/k
-module
def
L
×
is
naturally
isomorphic
to
the
G
K/k
-module
M
l
=
Hom(Q
l
/Z
l
,
μ
l
∞
)
⊗
Z
l
Q
l
[where
we
note
that
as
an
abstract
module,
M
l
is
isomorphic
to
Q
l
],
hence,
in
particular,
that
the
field
L
is
of
infinite
cardinality.
On
the
other
hand,
it
follows
from
elementary
Galois
theory
that
L
is
a
finite
Galois
extension
of
the
subfield
L
H
⊆
L
of
H-invariants
of
L.
Moreover,
since
H
acts
nontrivially
on
μ
l
,
hence
also
nontrivially
on
M
l
,
we
thus
conclude
—
from
the
corresponding
fact
for
the
action
×
of
nontrivial
subgroups
of
the
group
of
Teichmüller
representatives
[F
×
l
]
⊆
Z
l
on
Q
l
—
that
L
H
=
{0,
1}
is
a
set
of
cardinality
two,
hence
that
the
infinite
field
L
is
a
finite
field,
a
contradiction.
This
completes
the
proof
of
(v-a).
Note
that
(v-b)
if
it
was
indeed
the
case
that
I
O
×
admits
a
topological
field
structure
as
k
in
(v-a),
then
it
would
be
possible
to
consider
the
well-known
power
series
for
the
logarithm
[cf.
(iv-a),
(iv-b)].
Of
course,
it
follows
from
(v-a)
that
such
a
topological
field
structure
does
not
exist.
In
particular,
the
content
of
(v-a)
may
be
understood
as
pointing
roughly
in
the
same
direction
as
(iii-a),
(iii-b),
(iv-a),
(iv-b).
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
121
(vi)
Thus,
in
summary,
(vi-a)
the
fundamental
dichotomy,
in
the
context
of
the
p-adic
logarithm,
between
·
[finitely]
truncated
Kummer
theory
[as
in
(i-d)]
and
·
profinite
Kummer
theory
[as
in
(i-e)]
may
be
understood
in
terms
of
the
existence
[cf.
the
“k”
in
the
domain
of
log
k
in
(i-d)]
versus
non-existence
[cf.
the
“correspondence”
(Z,
ζ,
λ)
of
(iii-a)]
of
a
single
unified
basepoint
for
the
Kummer
theories
in
the
domain/codomain
of
log
k
or
Log
k
,
i.e.,
a
single
set
equipped
with
an
action
by
G
k
that
is
“sufficiently
rich”
as
to
admit
subquotients
[i.e.,
where
we
think
of
λ
as
in
(iii-a)
as
being
surjective]
that
correspond
to
the
Kummer
theories
in
the
domain/codomain
of
log
k
or
Log
k
.
That
is
to
say,
(vi-b)
in
the
case
of
profinite
Kummer
theory,
the
non-existence
of
such
a
single
unified
basepoint
means
that
one
must
treat
the
Kummer
theories
in
the
domain/codomain
of
Log
k
—
i.e.,
at
a
more
concrete
level,
the
sets
I
k
or
I
O
×
equipped
with
their
natural
multiplicative
structures,
profinite
k
cyclotomes,
and
G
k
-actions
—
as
being
independent
of
one
another.
In
particular,
it
follows,
essentially
formally,
from
(vi-b)
that
(vi-c)
one
must
think
of
the
copies
of
“G
k
”
that
appear
in
the
Kummer
theories
in
the
domain/codomain
of
the
p-adic
logarithm
—
which
may
in
fact
arise
as
quotients
of
copies
of
some
topological
group
Π
in
the
domain/codomain
of
the
p-adic
logarithm
—
as
being
related
to
one
another
not
as
groups
of
automorphisms
of
the
various
monoids
that
appear
in
the
Kummer
theories
in
the
domain/codomain
of
the
p-adic
logarithm,
but
rather
as
abstract
topological
groups,
i.e.,
which
may
be
related
to
one
another
only
by
means
of
some
∼
indeterminate
isomorphism
of
topological
groups
Π
→
Π
—
cf.
the
discussion
preceding
Example
3.8.1,
as
well
as
the
discus-
sion
following
Example
3.8.4,
concerning
the
necessity
of
working,
in
the
context
of
the
log-
and
Θ-links,
with
abstract
topological
groups,
that
is
to
say,
as
opposed
to
groups
of
ring/field
automorphisms,
i.e.,
“Galois
groups/arithmetic
fundamental
groups”.
Here,
we
observe
that
the
isomorphism
indeterminacy
discussed
in
(vi-c)
includes,
in
particular,
inner
automorphisms
of
Π.
This
chain
of
observations
(vi-a),
(vi-b),
(vi-c)
forms
the
starting
point
of
the
discussion
of
Example
3.8.4
below.
Example
3.8.4:
Symmetrizing
isomorphisms,
truncatibility,
and
the
log-
Kummer-
correspondence.
We
maintain
the
notation
of
Examples
3.8.2,
3.8.3.
122
SHINICHI
MOCHIZUKI
Also,
we
shall
write
“Out(−)”
for
the
group
of
outer
automorphisms
[i.e.,
arbi-
trary
automorphisms
considered
up
to
inner
automorphisms]
of
a
topological
group
“(−)”.
(i)
In
the
following
discussion,
we
consider
the
issue
of
compatibility
between
·
the
various
symmetrizing
isomorphisms
arising
from
the
action
of
the
±
F
l
-
and
F
l
-symmetries
on
[“abstract”]
copies
of
the
pair
G
k
O
k
[cf.
the
discussion
of
Examples
3.8.2,
3.8.3;
the
theory
of
[IUTchI],
§4,
§5,
§6;
[Alien],
§3.3,
(v);
[Alien],
§3.6,
(i),
(ii),
(iii)]
and
·
the
log-Kummer-correspondence
...
→
...
•
→
•
→
↓
•
→
...
...
◦
[cf.
[IUTchIII],
Theorem
3.11,
(ii)]
for
the
three
types
of
Kummer
theory,
namely,
(i-a)
the
Kummer
theory
for
[“abstract”]
copies
of
the
pair
G
k
O
,
which
is
k
based
on
the
classical
theory
of
Brauer
groups/local
class
field
theory
for
p-adic
local
fields
[cf.
[Alien],
§3.4,
(v)];
(i-b)
the
Kummer
theory
surrounding
theta
functions
and
theta
values
[cf.
[Alien],
§3.4,
(iii),
(iv);
[Alien],
§3.6,
(ii)];
(i-c)
the
Kummer
theory
surrounding
κ-coric
functions
and
copies
of
the
number
field
F
mod
[cf.
the
discussion
of
Example
3.8.2;
[IUTchI],
Defi-
nition
3.1,
(b);
[Alien],
§3.4,
(ii);
[Alien],
§3.6,
(iii)]]
that
appear
in
inter-universal
Teichmüller
theory
[cf.,
e.g.,
[Alien],
Fig.
3.10].
Be-
fore
proceeding,
we
recall
that
the
symmetrizing
isomorphisms
arising
from
the
±
action
of
the
F
l
-
and
F
l
-symmetries
on
[“abstract”]
copies
of
the
pair
G
k
O
k
differ
in
that
·
whereas
the
symmetrizing
isomorphisms
arising
from
the
F
±
l
-symmetries
are
free
of
inner
automorphism
indeterminacies
and
hence
give
rise
to
conjugate
synchronizations,
·
the
symmetrizing
isomorphisms
arising
from
the
F
l
-symmetries
neces-
sarily
involve
inner
automorphism
indeterminacies
[cf.
the
discussion
of
Example
3.8.2].
On
the
other
hand,
it
follows
immediately
—
i.e.,
by
considering
the
symmetrizing
isomorphisms
induced
by
arbitrary
elements
∼
of
Aut(X
K
)
→
Out(Π
X
K
)
[cf.
the
notation
of
[IUTchI],
Definition
3.1,
(d);
[Alien],
§3.3,
(v)]
—
that
if
one
forgets
about
the
issue
of
conjugate
synchronization
and
just
thinks
in
terms
of
arbitrary
[indeterminate]
isomorphisms
between
[“abstract”]
copies
of
the
pair
G
k
O
,
then
the
symmetrizing
isomorphisms
arising
k
from
the
action
of
the
F
-
and
F
±
l
l
-symmetries
on
[“abstract”]
copies
of
the
pair
G
k
O
in
fact
coincide.
k
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
123
(ii)
In
the
case
of
the
Kummer
theory
of
(i-a),
(ii-a)
compatibility
between
the
F
±
l
-symmetrizing
isomorphisms
—
i.e.,
without
conjugacy
indeterminacies!
[cf.
the
final
portion
of
(i)]
—
and
the
Kummer
theories
of
(i-a)
in
the
domain/codomain
of
the
log-
link
then
follows
formally
by
applying
transport
of
structure
via
the
F
±
l
-
symmetries
to
the
truncated
Kummer
theories
in
the
domain/codomain
of
the
log-link,
computed
relative
to
the
single
unified
basepoint
dis-
cussed
in
Example
3.8.3,
(i-d),
(vi-a),
at
each
evaluation
label
“t
∈
F
l
”
[cf.
the
discussion
of
[IUTchIII],
Remark
2.3.3,
(viii);
[Alien],
§3.6,
(ii)].
Here,
we
recall
that
(ii-b)
this
compatibility
is
necessary
to
define
the
diagonal
“0/
/
>”
—
i.e.,
with
respect
to
the
evaluation
labels
“t
∈
F
l
”
—
local
data
that
is
used
to
construct
the
local
unit
group
[i.e.,
“O
×μ
”]
portion
of
the
gluing
data
that
appears
in
the
Θ-link
[cf.
[IUTchIII],
Theorem
1.5,
(iii);
[IUTchIII],
Remark
1.5.1,
(i);
[Alien],
§3.6,
(ii)].
In
this
context,
it
is
important
to
recall
that
(ii-c)
this
local
unit
group
portion
of
the
Θ-link
gluing
data
satisfies
the
crucial
property
of
being
independent
of
the
evaluation
labels
“t
∈
F
l
”
—
which
are
only
well-defined
internally
within
a
particular
(Θ
±ell
NF-)Hodge
the-
ater!
—
hence
allows
one
to
construct
the
containers
[that
is
to
say,
in
the
form
of
tensor-packets
of
log-shells]
that
appear
in
the
multiradial
representation
of
the
Θ-pilot,
i.e.,
the
containers
that
make
it
possible
to
represent
the
Θ-pilot
in
the
domain
(Θ
±ell
NF-)Hodge
theater
of
the
Θ-link
in
terms
of
“external”
data
arising
from
the
codomain
(Θ
±ell
NF-
)Hodge
theater
of
the
Θ-link.
Put
another
way,
(ii-d)
if
it
were
the
case
that
the
containers
of
(ii-c)
could
only
be
constructed
from
the
domain
(Θ
±ell
NF-)Hodge
theater
of
the
Θ-link
in
a
way
that
involves
independent
conjugacy
indeterminacies
·
at
the
distinct
evaluation
labels
“t
∈
F
l
”
[cf.
(ii-b),
(ii-c)]
or
·
in
the
domain/codomain
of
the
log-link
[cf.
(ii-a)]
—
i.e.,
all
of
which
are
only
well-defined
internally
within
the
domain
(Θ
±ell
NF-)Hodge
theater
of
the
Θ-link!
—
then
it
would
follow
that
these
containers
cannot
be
constructed
in
a
way
that
is
well-defined
externally
to
the
domain
(Θ
±ell
NF-)Hodge
theater
of
the
Θ-link,
e.g.,
in
terms
of
data
arising
from
the
codomain
(Θ
±ell
NF-)Hodge
theater
of
the
Θ-link.
In
particular,
we
observe
that
the
truncatibility
of
the
Kummer
theory
of
(i-
a)
plays
a
fundamental
role
in
the
logical
structure
of
inter-universal
Teichmüller
theory.
(iii)
The
discussion
of
(ii-b),
(ii-c),
(ii-d)
centers
on
the
issue
of
synchro-
nizing
the
conjugacy
indeterminacies
at
the
diagonal
label
“
/
>”
in
the
do-
main/codomain
of
the
log-link.
This
focus
of
attention
on
the
diagonal
label
thus
prompts
the
following
question:
124
SHINICHI
MOCHIZUKI
(iii-a)
If
one
is
only
interested
in
synchronizing
the
conjugacy
indeterminacies
at
the
diagonal
label
“
/
>”
in
the
domain/codomain
of
the
log-link,
then
why
does
it
not
suffice
to
relate
the
profinite
[i.e.,
rather
than
truncated!]
versions
of
the
Kummer
theory
of
(i-a)
for
the
diagonal
label
“
/
>”
in
the
domain/codomain
of
the
log-link
via
a
single
[i.e.,
corresponding
to
∼
the
“single”
diagonal
label]
indeterminate
isomorphism
“Π
→
Π”
as
in
Example
3.8.3,
(vi-c)?
In
fact,
however,
the
approach
described
in
(iii-a)
is
not
sufficient
for
the
following
reason:
(iii-b)
Ultimately,
in
inter-universal
Teichmüller
theory,
one
is
interested
in
con-
structing
the
multiradial
representation
of
the
Θ-pilot
[cf.
[IUTchIII],
Theorem
3.11]
—
which
involves
the
theta
values
2
“q
j
”
v
at
v
∈
V
bad
—
i.e.,
whose
construction
depends,
in
an
essential
way,
on
the
use
of
[global]
independent
labels
“t
∈
F
l
”
[cf.
(ii-d)]
at
which
the
conjugacy
indeterminacies
are
nevertheless
synchronized
relative
to
a
single
basepoint
arising
from
the
global
additive
symmetry
portion
“D
±
”
of
the
(Θ
±ell
NF-)Hodge
theater
under
consideration
[cf.
[IUTchI],
Definition
6.1,
(v);
[IUTchII],
Corollary
4.5,
(iii),
(iv);
[IUTchII],
Corollary
4.6,
(iii),
(iv)].
That
is
to
say,
(iii-c)
the
requirements
discussed
in
(iii-b)
are
satisfied
by
the
approach
that
is
actually
taken
in
inter-universal
Teichmüller
theory
[cf.
(ii-a);
[IUTchIII],
Proposition
1.3,
(i);
[IUTchIII],
Remark
1.3.2],
i.e.,
of
synchronizing
the
conjugacy
indeterminacies
in
the
domain/codomain
of
the
log-link
locally
“label
by
label”,
for
the
various
labels
“t
∈
F
l
”
[cf.
(ii-d)],
so
that
arbitrary
conjugacy
indeterminacy
synchronizations
in
the
domain
of
the
log-link
are
reflected
faithfully
in
the
codomain
of
the
log-link.
On
the
other
hand,
(iii-d)
the
approach
discussed
in
(iii-c)
can
be
implemented
precisely
because
of
the
existence
of
the
canonical
single
unified
basepoints
for
the
do-
main/codomain
of
the
log-link
discussed
in
Example
3.8.3,
(vi-a),
i.e.,
which
are
available
only
in
the
case
of
truncated
Kummer
theory,
since
the
profinite
Kummer
theory
conjugacy
indeterminacies
of
Example
3.8.3,
(vi-c),
would
give
rise
[in
the
situation
of
the
approach
discussed
in
(iii-c)]
to
independent
conjugacy
indeterminacies
at
the
various
labels
“t
∈
F
l
”.
Thus,
in
summary,
the
discussion
of
the
present
(iii)
sheds
further
light
on
the
fundamental
role
played
by
the
truncatibility
of
the
Kummer
theory
of
(i-a)
in
the
logical
structure
of
inter-universal
Teichmüller
theory.
(iv)
In
the
case
of
the
Kummer
theory/Galois
evaluation
of
(i-c),
let
us
first
recall
from
the
discussion
of
[IUTchIII],
Remark
2.3.3,
(vi),
(vii)
[cf.
also
[Alien],
§3.4,
(ii)]
that
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
125
(iv-a)
the
fact
that
the
submonoid
I
ord
⊆
N
×
{±1}
generated
by
the
set
of
orders
of
the
zeroes/poles
[considered
as
signed
elements
of
N
×
{±1}]
of
the
rational
functions
that
appear
in
(i-c)
contains
—
i.e.,
unlike
the
case
with
the
theta
functions
that
appear
in
(i-b)!
—
elements
∈
{±1},
as
well
as
elements
∈
N,
means
that
the
cyclo-
tomic
rigidity
isomorphisms
obtained
in
(i-c)
may
only
be
constructed
in
a
fashion
consistent
with
the
anabelian
reconstruction
algorithms
of
[AbsTopIII],
Theorem
1.9
[cf.
also
[IUTchI],
Remark
3.1.2,
(ii),
(iii)]
if
one
constructs
these
cyclotomic
rigidity
isomorphisms
·
via
profinite
Kummer
theory
[i.e.,
by
applying
the
fact
that
×
=
{1}]
and
Q
>0
∩
Z
·
up
to
an
indeterminacy
given
by
multiplication
by
elements
ord
{±1}
to
the
of
the
image
I
ord
±
=
{±1}
of
the
projection
I
second
factor.
—
i.e.,
in
particular,
relative
to
the
constraints
discussed
in
Example
3.8.3,
(vi-b),
(vi-c).
That
is
to
say,
it
follows
from
the
discussion
of
Example
3.8.3,
(vi-b),
(vi-c)
—
cf.
also
·
the
splitting/decoupling
of
the
unit
group
portion
from
the
pseudo-
monoid
of
κ-coric
rational
functions
[as
discussed
in
[IUTchI],
Example
5.1,
(v);
[Alien],
§3.4,
(ii)];
·
the
non-interference
properties
satisfied
by
the
Frobenius-like
copies
of
×
in
the
domain/codomain
of
the
log-link
[as
discussed
in
[IUTchIII],
F
mod
Proposition
3.10,
(ii);
[Alien],
§3.7,
(i)]
—
that
(iv-b)
the
Kummer
theories/Galois
evaluation
operations
of
(i-c)
in
the
do-
main/codomain
of
the
log-link
·
involve
[pseudo-]monoids
that
must
be
treated
indepen-
dently
of
one
another,
hence,
in
particular,
·
may
be
related
to
one
another
only
up
to
indeterminacies
that
involve
indeterminate
isomorphisms
between
corresponding
Ga-
lois
groups/arithmetic
fundamental
groups
in
the
domain/codomain
of
the
log-link
[cf.,
especially,
the
discussion
of
the
final
portion
of
Example
3.8.3,
(vi)].
On
the
other
hand,
(iv-c)
the
compatibility
between
the
F
l
-symmetrizing
isomorphisms
—
i.e.,
with
conjugacy
indeterminacies!
[cf.
the
discussion
of
the
final
portion
of
(i)]
—
and
the
Kummer
theories/Galois
evaluation
operations
of
(i-c)
in
the
domain/codomain
of
the
log-link
follows
formally
by
applying
transport
of
structure
via
the
F
l
-symmetries
to
the
profinite
Kummer
126
SHINICHI
MOCHIZUKI
theories
of
(i-c)
in
the
domain/codomain
of
the
log-link,
together
with
the
compatibility
of
the
log-link
with
these
F
l
-symmetrizing
isomorphisms
[cf.
[IUTchIII],
Proposition
1.3,
(i),
(ii);
[IUTchIII],
Remark
1.3.3,
(ii)].
As
a
result
of
the
various
indeterminacies
of
(iv-b),
(iv-c),
(iv-d)
in
the
case
of
the
Kummer
theories/Galois
evaluation
operations
of
(i-
c),
the
only
diagonal
“0/
/
>”
—
i.e.,
with
respect
to
the
evaluation
labels
“j
∈
F
l
”
—
number
field
inside
the
algebraic
closure
F
that
is
well-defined
externally
to
the
domain
(Θ
±ell
NF-)Hodge
theater
of
the
Θ-link
[cf.
the
discussion
of
the
multiradial
representation
in
(ii-c),
(ii-d)]
is
F
mod
[cf.
[Alien],
§3.6,
(iii)].
Here,
we
observe
that
(iv-e)
the
I
ord
±
-indeterminacies
of
(iv-a)
may
be
synchronized
·
relative
to
the
F
l
-symmetry
by
applying
the
“linear
dis-
jointness”
of
the
theory
of
κ-coric
functions
from
SL
2
(F
l
)
[cf.
the
second
display
of
[Alien],
§3.6,
(iii)];
·
relative
to
the
distinct
valuations
v
∈
V
in
local-global
com-
parisons
of
the
Kummer
theories/Galois
evaluation
operations
of
(i-c)
by
means
of
comparison
with
the
cyclotomic
rigidity
iso-
morphisms
arising
from
the
Kummer
theories
of
(i-a)
[i.e.,
which
are
not
subject
to
I
ord
±
-indeterminacies]
[cf.
the
discussion
of
the
final
portion
of
[IUTchIII],
Remark
2.3.3,
(vi)],
but
·
not
relative
to
the
domain/codomain
of
the
log-link
since
the
relevant
[Frobenius-like]
monoids
in
the
domain/codomain
of
the
log-link
are
not
set-theoretically
related
to
one
another
via
the
log-link
[cf.
(iv-b)].
In
particular,
we
note
that
one
fundamental
aspect
of
the
Kummer
theory/Galois
evaluation
of
(i-c)
that
underlies
the
compatibility
and
synchronization
properties
of
(iv-c),
(iv-e)
is
the
F
l
-symmetricity
of
the
Kummer
theory/Galois
evaluation
of
(i-c)
[cf.
the
discussion
of
[IUTchII],
Remark
1.1.1,
(v);
the
first
display
of
[IUTchIII],
Remark
2.3.3,
(iii);
the
discussion
of
[Alien],
§3.6,
(iii)].
(v)
The
Kummer
theory/Galois
evaluation
of
(i-b),
which
does
not
satisfy
the
condition
of
F
±
l
-symmetricity,
exhibits
qualitatively
fundamentally
different
behavior
from
the
F
l
-symmetric
Kummer
theory/Galois
evaluation
of
(i-c)
[cf.
the
discussion
of
[IUTchII],
Remark
1.1.1,
(v);
the
first
display
of
[IUTchIII],
Remark
2.3.3,
(iii)].
If,
for
instance,
(v-a)
one
attempts
to
apply
the
approach
via
profinite
Kummer
theory
of
(iv-b),
(iv-c)
to
the
task
of
verifying
some
sort
of
compatibility
be-
tween
the
F
±
l
-symmetrizing
isomorphisms
—
i.e.,
with
conjugacy
indeterminacies
arising
from
log-link
domain/codomain
comparisons,
as
discussed
in
Example
3.8.3,
(vi-b),
(vi-c)
[cf.
also
the
discussion
of
(i),
(ii),
(iii),
especially
(iii-d),
of
the
present
Example
3.8.4]!
—
and
the
Kummer
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
theories/Galois
evaluation
operations
of
(i-b)
in
the
domain/codomain
of
the
log-link,
then
one
encounters
the
following
situation:
(v-b)
In
order
to
compare,
via
the
F
±
l
-symmetry,
the
Kummer
theories/Galois
evaluation
operations
at
different
t
∈
F
l
[i.e.,
for
distinct
theta
values
2
“q
j
”,
where
v
∈
V
bad
],
it
is
necessary
to
establish
a(n)
—
a
priori
self-
v
contradictory!—
situation
in
which
the
F
±
l
-symmetry
permutes
the
labels
“t
∈
F
l
”
in
a
nontrivial
fashion,
but
acts
trivially
on
the
[non-
F
±
l
-symmetric!]
étale
theta
function!
Here,
we
recall
[cf.
the
theory
of
[IUTchII],
§2]
that
(v-c)
in
the
context
of
Galois
evaluation,
the
étale
theta
function
may
be
thought
of
as
a
cohomology
class
some
closed
subgroup
some
cyclotome
associated
to
the
∈
H
1
,
of
Π
v
Galois
evaluation
label
t
∈
F
l
—
that
is
to
say,
where
[cf.
(v-b)!]
the
F
±
l
-symmetry
is
to
act
nontriv-
ially
on
the
Galois
evaluation
cyclotome
[i.e.,
the
second
argument
of
“H
1
(−)”],
but
trivially
on
the
closed
subgroup
of
Π
v
[i.e.,
the
first
argument
of
“H
1
(−)”]!
Before
proceeding,
it
is
also
of
interest
to
recall
that
(v-d)
a
profinite
Kummer
theory
situation
of
the
sort
discussed
in
(v-a),
(v-b),
(v-c)
occurs,
for
instance,
if
one
replaces
the
theta
functions
that
occur
in
inter-universal
Teichmüller
theory
by
their
N
-th
powers,
where
N
is
an
integer
≥
2
[cf.
the
discussion
of
[IUTchIII],
Remark
2.3.3,
(vi),
(vii)].
Now
returning
to
the
discussion
of
(v-b),
(v-c),
we
observe
that
(v-e)
it
is
essentially
a
tautology
that
the
only
“solution”
to
the
[a
priori]
self-contradictory
simultaneous
“trivial/nontrivial
action”
condi-
tion
“(SimCon)”
of
(v-b),
(v-c)
lies
in
working
with
objects
that
are
invariant
with
respect
to
the
F
±
l
-symmetry,
i.e.,
at
a
more
concrete
level,
with
a
single
unified
set-theoretic
basepoint
—
which
allows
one
to
compute
the
various
F
±
l
-symmetrizing
isomor-
phisms
by
projecting
to
the
single
copy
of
“G
v
”
determined
by
the
base-
point.
On
the
other
hand,
(v-f)
the
existence
of
the
conjugacy
indeterminacies
inherent
in
the
profi-
nite
Kummer
theory
F
±
l
-symmetrizing
isomorphisms
of
(v-a)
means
that
any
“single
unified
basepoint”
as
in
(v-e)
is
well-defined
only
up
to
conju-
gacy
indeterminacies
—
a
situation
that
is
unacceptable
in
inter-universal
Teichmüller
theory
since
it
would
mean
that
the
coefficient
cyclotomes
in
×
-multiples,
i.e.,
that
the
étale
(v-c)
are
well-defined
only
up
to
certain
Z
127
128
SHINICHI
MOCHIZUKI
2
theta
functions
and
theta
values
[i.e.,
“q
j
”,
where
v
∈
V
bad
]
in
the
v
×
-powers
[cf.
the
discussion
theory
are
well-defined
only
up
to
certain
Z
of
[IUTchIII],
Remark
2.1.1,
(v)]!
That
is
to
say,
(v-g)
the
only
way
to
avoid
the
pathologies
of
(v-f)
is
to
replace
the
profinite
Kummer
theory
in
the
discussion
of
(v-a),
(v-b),
(v-c),
(v-e),
(v-f)
by
the
truncated
Kummer
theory
of
(i-b)
[cf.
the
discussion
of
the
final
por-
tion
of
[Alien],
§3.6,
(ii)],
so
that
we
can
apply
the
single
unified
basepoint
of
Example
3.8.3,
(i-d),
(vi-a)
[cf.
also
the
discussion
of
(ii)
in
the
present
Example
3.8.4]
to
obtain
a
single
unified
set-theoretic
basepoint
as
in
(v-e),
but
which
is
well-defined
up
to
geometric
fundamental
group
conjugacy
indeterminacies
[i.e.,
indeterminacies
arising
from
conjugation
by
elements
of
the
geometric
fundametric
fundamental
groups
involved],
which
are,
at
any
rate,
inherent
in
the
F
±
l
-symmetrizing
iso-
morphisms
and,
moreover,
[unlike
the
situation
discussed
in
(v-f)!]
do
not
have
any
effect
on
the
computation
of
these
F
±
l
-symmetrizing
iso-
morphisms
by
projecting
to
the
single
copy
of
“G
v
”
determined
by
the
basepoint
—
cf.
[IUTchII],
Remark
1.1.1,
(iv),
(v);
[IUTchII],
Remark
2.6.1,
(i),
(ii).
Before
proceeding,
it
is
interesting
to
observe
that
(v-h)
the
“tautological
resolution”
of
the
[a
priori]
self-contradictory
simul-
taneity
condition
(SimCon)
discussed
in
(v-e),
(v-g)
by
considering
“invariants”
—
i.e.,
in
the
situation
of
(v-e),
(v-g),
a
single
unified
set-
theoretic
basepoint
—
is
formally
highly
reminiscent
of
the
“tautological
resolution”
of
the
log-shifts
in
the
left-
and
right-hand
columns
of
the
“infinite
H”
of
(InfH)
by
means
of
the
construction
of
invariants
[cf.
the
discussion
surrounding
(logORInd),
(Di/NDi)
in
§3.11
below;
the
discus-
sion
of
(Stp7)
in
§3.10
below].
Thus,
in
summary,
the
truncatibility
of
the
Kummer
theory/Galois
evaluation
operations
of
(i-b)
[cf.
the
discussion
of
[IUTchIII],
Remark
2.3.3,
(vi),
(vii),
(viii);
the
final
portion
of
[Alien],
§3.6,
(ii)]
plays
a
fundamental
role
in
the
logical
structure
of
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
Example
3.3.2,
(vii)].
(vi)
Recall
that
the
single
unified
basepoint
of
(v-g)
is
completely
deter-
mined
by
the
connected
subgraph
“Γ
X
⊆
Γ
X
”,
or,
equivalently,
the
connected
⊆
Γ
Ÿ
”
[cf.
[IUTchII],
Remark
2.6.1,
(i),
(ii);
[IUTchII],
Remark
subgraph
“Γ
Ÿ
2.6.3,
(i)].
Here,
we
recall
further
from
[IUTchII],
Remark
2.6.3,
(i),
that
Γ
X
or
Γ
Ÿ
may
be
thought
of
as
a
“copy
Γ
of
the
real
line
R”,
in
which
the
integers
Z
⊆
R
are
taken
to
be
the
vertices,
and
the
line
segments
joining
the
integers
are
taken
to
be
the
edges.
In
light
of
the
central
role
played
by
the
singled
unified
basepoint
of
(v-g)
in
the
discussion
of
(v)
—
and
indeed
in
the
entire
logical
structure
of
inter-universal
Teichmüller
theory!
—
it
is
of
interest
to
recall
from
the
discus-
sion
of
[IUTchII],
Remark
2.6.3,
that
this
subgraph
“Γ
”
is
[essentially]
uniquely
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
129
determined
by
various
natural
conditions,
which
must
be
satisfied
in
order
for
the
theory
to
operate
in
the
desired
fashion.
Indeed,
let
us
first
recall
from
the
discussion
of
[IUTchII],
Remark
2.5.2,
(i),
(ii),
(iii),
(iv);
[IUTchII],
Remark
2.6.3,
(ii),
that
(vi-a)
the
various
geometric
fundamental
groups
“Δ”
that
appear
act
on
the
various
vertices
t
∈
{−l
,
.
.
.
,
−1,
0,
1,
.
.
.
,
l
}
⊆
Γ
independently
—
where
we
recall
that
l
=
l−1
2
,
and
that
(vi-b)
the
actions
referred
to
in
(vi-a)
of
geometric
fundamental
groups
“Δ”
on
the
“copy
Γ
of
the
real
line
R”
amount
to
the
conventional
action
of
def
G
Γ
=
l
·
Z
{±1}
[i.e.,
the
group
generated
by
translations
by
elements
of
l
·
Z
and
multiplication
by
−1]
on
the
real
line
R.
def
∩
Z;
Write
Γ
Z
=
Γ
def
±Γ
Z
=
Γ
Z
∪
−Γ
Z
⊆
R.
Next,
recall
that
(vi-c)
since
the
vertex
0
∈
Γ
is
the
unique
vertex
in
Γ
that
may
be
used
to
define
the
crucial
splittings
of
theta
monoids
discussed
in
[IUTchII],
Corollary
2.6,
(ii),
it
is
necessary
that
0
∈
Γ
Z
[cf.
[IUTchII],
Remark
2.6.3,
(i)].
On
the
other
hand,
(vi-c),
together
with
the
existence
of
the
independent
G
Γ
-
indeterminacies
of
(vi-a),
(vi-b),
imply
that
(vi-d)
if
Γ
contains
a
connected
component
Γ
∗
that
is
not
contained
in
l
·
Z
(⊆
Γ),
then
any
theta
value
2
q
j
v
obtained
via
Galois
evaluation
for
j
∈
Γ
∗
∩
Z
⊆
Γ,
is
well-defined
only
up
to
a
G
Γ
-indeterminacy,
i.e.,
up
to
a
possible
confusion
between
j
and
G
Γ
-translates
j
of
j.
(vi-e)
any
theta
value
q
j
2
v
obtained
via
Galois
evaluation
for
j
∈
Γ
Z
,
is
well-defined
only
up
to
,
i.e.,
up
to
a
possible
confusion
between
j
a
G
Γ
-indeterminacy
within
Γ
Z
and
G
Γ
-translates
j
of
j
that
lie
inside
Γ
Z
.
Here,
it
is
important
to
recall
[cf.
[IUTchI],
Remark
3.5.1,
(ii);
[IUTchII],
Remark
2.6.3,
(iv);
[IUTchII],
Corollary
4.5,
(v);
[IUTchII],
Corollary
4.6,
(v)]
that
(vi-f)
any
indeterminacies
concerning
“j”,
“j
”
of
the
sort
described
in
(vi-d),
(vi-e)
for
nonzero
j,
j
∈
Z
with
distinct
absolute
values
would
result
in
indeterminacies
in
the
“vector
of
ratios”
[cf.
[IUTchII],
Corollary
4.5,
(v)]
that
determines
the
structure
of
the
global
realified
Gaussian
Frobenioids,
i.e.,
would
result
in
violations
of
the
global
product
formula
relating
the
value
groups
at
different
v
∈
V.
In
particular,
(vi-c),
(vi-d),
(vi-e),
(vi-f)
imply
that
(vi-g)
the
natural
map
±Γ
Z
/{±1}
→
Z/G
Γ
130
SHINICHI
MOCHIZUKI
—
i.e.,
from
the
set
of
{±1}-orbits
of
±Γ
Z
to
the
set
of
G
Γ
-orbits
in
Z
—
is
injective
[cf.
[IUTchII],
Remark
2.6.3,
(ii)];
(vi-h)
the
subgraph
Γ
⊆
Γ
is
connected
[cf.
[IUTchII],
Remark
2.6.3,
(i)].
Now
one
verifies
immediately
that
(vi-c),
(vi-g),
(vi-h)
imply
formally
that
(vi-i)
Γ
⊆
Γ
is
a
connected
subgraph
that
contains
0
and
is
contained
in
the
closed
interval
[−l
,
l
].
Next,
we
observe
that,
in
light
of
(vi-e),
(vi-i),
(vi-j)
replacing
Γ
Z
by
±Γ
Z
does
not
have
any
effect
on
the
validity
of
(vi-i)
or
on
the
way
in
which
the
subgraph
Γ
Z
is
used
in
inter-universal
Teichmüller
theory;
in
particular,
we
may
assume,
without
loss
of
generality,
that
the
symmetry
condition
Γ
Z
=
±Γ
Z
holds.
On
the
other
hand,
(vi-k)
the
estimates
that
are
ultimately
obtained
in
inter-universal
Teichmüller
theory
[cf.
[IUTchIII],
Corollary
3.12;
[IUTchIV],
Theorem
1.10]
are
op-
timized
precisely
when
the
average
of
the
squares
j
2
of
the
nonzero
ele-
ments
j
∈
Γ
Z
=
±Γ
Z
is
maximized
[cf.
[IUTchII],
Remark
2.6.3,
(ii)],
i.e.,
when
Γ
=
±Γ
Z
is
given
precisely
by
the
closed
interval
[−l
,
l
]
[which
implies
that
the
natural
map
of
(vi-g)
is
bijective].
That
is
to
say,
in
summary,
when
subject
to
the
symmetry
condition
Γ
Z
=
±Γ
Z
of
(vi-j)
and
the
optimization
condition
of
(vi-k),
the
subgraph
Γ
⊆
Γ
is
in
fact
uniquely
determined
[cf.
[IUTchII],
Remark
2.6.3,
(v)].
Unlike
the
situations
considered
in
[SGA1]
[cf.
the
discussion
of
Example
3.8.1],
in
which
the
ring/scheme
structures
of
the
various
distinct
schemes
that
appear
are
coric,
the
ring
structures
of
the
rings
that
appear
on
either
side
of
the
log-
and
Θ-links
of
inter-universal
Teichmüller
theory
—
i.e.,
such
as
number
fields
or
completions
of
number
fields
at
various
valuations
—
are
not
coric
with
respect
to
the
respective
links.
This
leads
one
naturally
to
consider
weaker
structures
[cf.
the
discussion
of
Example
3.2.2,
(i),
(ii),
(iv)]
the
discussion
of
Example
3.8.2,
(iii),
(iv);
the
discussion
of
Example
3.8.3,
(vi)]
such
as
·
abstract
topological
groups,
in
the
case
of
the
profinite
Kummer
theories
in
the
domain/codomain
of
the
log-link,
or
·
sets
equipped
with
a
topology
and
a
continuous
action
of
a
topological
group,
in
the
case
of
the
log-link,
or
·
realified
Frobenioids
[in
the
sense
of
[IUTchIII],
Theorem
1.5,
(v)]
or
topological
monoids
equipped
with
a
continuous
action
of
a
topological
group,
in
the
case
of
the
Θ-link,
which
are
indeed
coric
with
respect
to
the
respective
links.
Indeed,
it
is
pre-
cisely
this
sort
of
consideration
—
i.e.,
of
weaker
coric
structures
to
relate
the
uni-
verses/Galois
categories/étale
fundamental
groups
associated
to
ring/scheme
struc-
tures
on
opposite
sides
of
the
links
under
consideration
[cf.
the
discussion
preceding
Example
3.8.1]
—
that
gave
rise
to
the
term
“inter-universal”.
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
131
Here,
we
note
that
it
is
of
fundamental
importance
that
these
topological
groups
[which
typically
in
fact
arise
as
Galois
groups
or
arithmetic
fundamental
groups
of
schemes]
be
treated
as
abstract
topological
groups,
rather
than
as
Galois
groups
or
arithmetic
fundamental
groups
[cf.
the
discussion
at
the
beginning
of
§3.2;
the
discussion
of
Example
3.8.2,
(iii),
(iv);
the
discussion
of
Example
3.8.3,
(vi)].
That
is
to
say,
to
treat
these
topological
groups
as
Galois
groups
or
arithmetic
fundamental
groups
requires
the
use
of
the
ring/scheme
structures
involved,
i.e.,
the
use
of
structures
which
are
not
available
since
they
are
not
common/coric
to
the
rings/schemes
that
appear
on
opposite
sides
of
the
log-/Θ-link
[cf.
the
discussion
of
[Alien],
§2.10;
[IUTchIII],
Remarks
1.1.2,
1.2.4,
1.2.5;
[IUTchIV],
Remarks
3.6.1,
3.6.2,
3.6.3].
In
this
context,
it
is
also
of
fundamental
importance
to
observe
that
it
is
precisely
because
these
topological
groups
must
be
treated
as
abstract
topological
groups
that
anabelian
results
play
a
central
role
in
inter-universal
Teichmüller
theory.
One
consequence
of
the
constraint
[discussed
above]
that
one
must
typically
work,
in
inter-universal
Teichmüller
theory,
with
structures
that
are
substantially
weaker
than
ring
structures
is
the
necessity,
in
inter-universal
Teichmüller
theory,
of
allowing
for
various
indeterminacies,
such
as
(Ind1),
(Ind2),
(Ind3),
that
are
somewhat
more
involved
than
the
relatively
simple
inner
automorphism
indetermi-
nacies
that
occur
in
[SGA1].
Here,
we
recall
that
from
the
discussion
of
(∧(∨)-Chn)
in
§3.7
that
it
is
precisely
the
numerous
indeterminacies
that
arise
in
inter-universal
Teichmüller
theory
that
give
rise
to
the
numerous
logical
OR
relations
“∨”
in
the
display
of
(∧(∨)-Chn).
On
the
other
hand,
once
one
takes
such
indeterminacies
into
account,
i.e.,
once
one
consents
to
work
with
various
objects
“up
to
certain
suitable
indeterminacies”
—
e.g.,
by
means
of
poly-morphisms,
as
discussed
in
§3.7
—
it
is
natural
to
identify,
by
applying
(ExtInd2)
[as
discussed
in
§3.6],
objects
that
are
related
to
one
another
by
means
of
collections
of
isomorphisms
[i.e.,
poly-isomorphisms]
that
are
uniquely
determined
up
to
suitable
indeterminacies.
Here,
we
observe
that
this
sort
of
(ExtInd2)
identification
that
occurs
re-
peatedly
in
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
§3.6]
may
at
first
glance
appear
somewhat
novel.
In
fact,
however,
from
the
point
of
view
of
mathematical
foundations
—
i.e.,
just
as
in
the
discussion
of
inter-universality
given
above!
—
this
sort
of
(ExtInd2)
identification
is
qualitatively
very
similar
to
nu-
merous
classical
constructions
such
as
the
following:
(AlgCl)
the
notion
of
an
algebraic
closure
of
a
field
[cf.
the
discussion
of
Example
3.8.1],
which
is
not
constrained
to
be
a
specific
set
constructed
from
the
field;
(DrInv)
various
categorical
constructions
such
as
direct
and
inverse
limits
[i.e.,
such
as
fiber
products
of
schemes]
that
are
defined
by
means
of
some
sort
of
universal
property,
and
which
are
not
constrained
to
be
specific
sets
even
when
the
given
direct
or
inverse
systems
are
specified
set-theoretically;
132
SHINICHI
MOCHIZUKI
(HomRs)
various
constructions
of
(co)homology
modules
in
homological
al-
gebra
that
depend
on
the
use
of
resolutions
that
satisfy
certain
abstract
properties,
but
which
are
not
constrained
in
a
strict
set-theoretic
sense
even
if
the
original
objects
resolved
by
such
resolutions
are
specified
set-
theoretically.
That
is
to
say,
in
each
of
the
classical
constructions,
the
“output
object”
is,
strictly
speaking,
from
the
point
of
view
of
mathematical
foundations,
not
well-defined
as
a
particular
set,
but
rather
as
a
collection
of
sets
[where
we
note
that,
typically,
this
“collection”
is
not
a
set!]
that
are
related
to
one
another
—
and
hence,
in
common
practice,
identified
with
one
another,
in
the
fashion
of
(ExtInd2)!
—
via
unique
[modulo,
say,
some
sort
of
well-defined
indeterminacy]
isomorphisms
by
means
of
some
sort
of
“universal”
property.
In
this
context,
it
is
also
important
to
note
that,
from
a
foundational
point
of
view,
the
sort
of
“(sub)quotient”
obtained
by
applying
(ExtInd2)
[cf.
the
discussion
of
“(sub)quotients”
in
(sQLTL)
and
indeed
throughout
§3.6]
must
be
regarded,
a
priori,
as
a
formal
(sub)quotient,
i.e.,
as
some
sort
of
diagram
of
arrows.
That
is
to
say,
at
least
from
an
a
priori
point
of
view,
(NSsQ)
any
explicit
construction
of
a
“naive
set-theoretic
(sub)quotient”
necessarily
requires
the
use
of
some
sort
of
set-theoretic
enumeration
of
each
of
the
individual
[set-theoretic]
objects
that
are
identified,
up
to
isomorphism,
via
an
application
of
(ExtInd2).
On
the
other
hand,
as
is
well-known,
typically
such
set-theoretic
enumerations
—
which
often
reduce,
roughly
speaking,
to
consideration
of
the
“set
of
all
sets”!
—
lead
immediately
to
a
contradiction.
Indeed,
it
is
precisely
this
aspect
of
the
constructions
of
inter-universal
Teichmüller
theory
that
motivated
the
author
to
include
the
discussion
of
species
in
[IUTchIV],
§3.
Finally,
we
recall
—
cf.
also
the
discussion
of
§3.10
[especially,
(Stp7)]
below
—
that
(LVsQ)
it
is
only
in
the
final
portion
of
inter-universal
Teichmüller
theory,
i.e.,
once
one
obtains
a
formal
(sub)quotient
that
forms
a
“closed
loop”,
that
one
may
pass
from
this
formal
(sub)quotient
to
a
“coarse/set-theoretic
(sub)quotient”
by
taking
the
log-volume
[cf.
the
discussion
of
[Alien],
§3.11,
(v);
[IUTchIII],
Remark
3.9.5,
(ix);
Steps
(x),
(xi)
of
the
proof
of
[IUTchIII],
Corollary
3.12].
§3.9.
Passage
and
descent
to
underlying
structures
One
fundamental
aspect
of
inter-universal
Teichmüller
theory
lies
in
the
use
of
numerous
functorial
algorithms
that
consist
of
the
construction
input
data
output
data
of
certain
output
data
associated
to
given
input
data.
When
one
applies
such
func-
torial
algorithms,
there
are
two
ways
in
which
the
output
data
may
be
treated
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
133
[cf.
[Alien],
§2.7,
(iii);
the
discussion
of
“post-anabelian
structures”
in
[IUTchII],
Remark
1.11.3,
(iii),
(v);
[IUTchIII],
Remark
1.2.2,
(vii)]
:
(UdOut)
One
may
consider
the
output
data
independently
of
the
given
input
data
and
functorial
algorithms
used
to
construct
the
output
data.
In
this
case,
the
output
data
may
be
regarded
as
a
sort
of
“underlying
structure”
associated
to
the
input
data.
(InOut)
One
may
consider
the
output
data
as
data
equipped
with
the
additional
structure
constituted
by
the
input
data,
together
with
the
functorial
algo-
rithm
that
gave
rise
to
the
output
data
by
applying
the
algorithm
to
the
input
data.
Typical
examples
of
this
phenomenon
in
inter-universal
Teichmüller
theory
are
the
following
[cf.
the
notational
conventions
of
[IUTchI],
Definition
3.1,
(e),
(f)]:
(sQGOut)
Functorial
algorithms
that
associate
to
Π
v
[where
v
∈
V
non
]
some
sub-
quotient
group
of
Π
v
,
such
as,
for
instance,
the
quotient
Π
v
G
v
:
In
this
sort
of
situation,
treatment
of
the
output
data
[i.e.,
subquotient
group
of
Π
v
]
according
to
(InOut)
is
indicated
by
a
“(Π
v
)”
following
the
notation
for
the
particular
subquotient
under
consideration;
by
contrast,
treatment
of
the
output
data
[i.e.,
subquotient
group
of
Π
v
]
according
to
(UdOut)
is
indicated
by
the
omission
of
this
“(Π
v
)”.
(MnOut)
Functorial
algorithms
that
associate
to
Π
v
[where
v
∈
V
non
]
some
sort
of
[abelian]
monoid
equipped
with
a
continuous
action
by
Π
v
,
such
as,
for
instance,
[data
isomorphic
to]
various
subquotient
monoids
[i.e.,
“O
”,
×
“O
×
”,
“O
×μ
”,
etc.]
of
the
multiplicative
monoid
F
v
:
In
this
sort
of
situation,
treatment
of
the
output
data
[i.e.,
monoid
equipped
with
an
action
by
Π
v
]
according
to
(InOut)
is
indicated
by
a
“(Π
v
)”
following
the
notation
for
the
particular
monoid
equipped
with
an
action
by
Π
v
under
consideration;
by
contrast,
treatment
of
the
output
data
[i.e.,
monoid
equipped
with
an
action
by
Π
v
]
according
to
(UdOut)
is
indicated
by
the
omission
of
this
“(Π
v
)”.
(PSOut)
Functorial
algorithms
that
associate
some
sort
of
prime-strip
to
some
sort
of
input
data:
In
this
sort
of
situation,
treatment
of
the
output
data
[i.e.,
some
sort
of
prime-strip]
according
to
(InOut)
is
indicated
by
a
“(−)”
[where
“−”
is
the
given
input
data]
following
the
notation
for
the
particular
prime-strip
under
consideration;
by
contrast,
treatment
of
the
output
data
[i.e.,
some
sort
of
prime-strip]
according
to
(UdOut)
is
indicated
by
the
omission
of
this
“(−)”.
Perhaps
the
most
central
example
of
(PSOut)
in
inter-universal
Teichmüller
theory
is
the
notion
of
the
“q-/Θ-intertwinings”
on
an
F
×μ
-prime-strip
[cf.
the
discussion
of
[Alien],
§3.11,
(v);
[IUTchIII],
Remark
3.9.5,
(viii),
(ix);
[IUTchIII],
Remark
3.12.2,
(ii)]:
(ItwOut)
This
terminology
refers
to
the
treatment
of
the
F
×μ
-prime-strip
ac-
cording
to
(InOut),
relative
to
the
functorial
algorithm
for
constructing
the
q-pilot
F
×μ
-prime-strip
[in
the
case
of
the
“q-intertwining”]
or
the
Θ-pilot
F
×μ
-prime-strip
[in
the
case
of
the
“Θ-intertwining”]
from
some
Θ
±ell
NF-
or
D-Θ
±ell
NF-Hodge
theater.
134
SHINICHI
MOCHIZUKI
In
any
situation
in
which
one
considers
a
construction
from
the
point
of
view
of
(UdOut)
—
that
is
to
say,
as
a
construction
that
produces
“underlying
data”
[i.e.,
“output
data”]
from
“original
data”
[i.e.,
“input
data”]
input
data
output
data
||
||
original
data
underlying
data
—
it
is
natural
to
consider
the
issue
of
descent
to
[a
functorial
algorithm
in]
the
underlying
data
of
a
functorial
algorithm
in
the
original
data.
Here,
we
say
that
a
functorial
algorithm
Φ
in
the
original
data
descends
to
a
functorial
algorithm
Ψ
in
the
underlying
data
if
there
exists
a
functorial
isomorphism
∼
→
Φ
Ψ|
original
data
between
Φ
and
the
restriction
of
Ψ,
i.e.,
relative
to
the
given
construction
original
data
underlying
data.
That
is
to
say,
roughly
speaking,
to
say
that
the
functorial
algorithm
Φ
in
the
original
data
descends
to
the
underlying
data
means,
in
essence,
that
although
the
construction
constituted
by
Φ
depends,
a
priori,
on
the
“finer”
original
data,
in
fact,
up
to
natural
isomorphism,
it
only
depends
on
the
“coarser”
underlying
data.
One
elementary
example
of
the
phenomenon
of
descent
may
be
seen
in
the
situation
discussed
in
(HomRs)
in
§3.8:
(HmDsc)
The
various
constructions
of
(co)homology
modules
in
homological
algebra
are,
strictly
speaking,
constructions
that
require
as
input
data
not
just
some
given
module
[whose
(co)homology
is
computed
by
the
construc-
tion],
but
also
some
sort
of
resolution
of
the
given
module
that
satisfies
certain
properties.
In
fact,
however,
such
constructions
of
(co)homology
modules
typically
descend,
up
to
unique
isomorphism,
to
constructions
whose
input
data
consists
solely
of
the
given
module.
Another
illustrative
elementary
example
of
the
phenomenon
of
descent
is
the
fol-
lowing:
Example
3.9.1:
Categories
of
open
subschemes.
topological
space.
Write
Let
X
be
a
scheme,
T
a
·
|X|
for
the
underlying
topological
space
of
X,
·
Open(X)
for
the
category
of
open
subschemes
of
X
and
open
immersions
over
X,
·
Open(T
)
for
the
category
of
open
subsets
of
T
and
open
immersions
over
T
.
Then
the
functorial
algorithm
X
→
Open(X)
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
135
—
defined,
say,
on
the
category
of
schemes
and
morphisms
of
schemes
—
is
eas-
ily
verified
to
descend,
relative
to
the
construction
X
|X|,
to
the
functorial
algorithm
T
→
Open(T
)
—
defined,
say,
on
the
category
of
topological
spaces
and
continuous
maps
of
topo-
logical
spaces.
That
is
to
say,
one
verifies
immediately
that
there
is
a
natural
functorial
isomorphism
Open(X)
∼
→
Open(|X|)
[i.e.,
in
this
case,
following
the
conventions
employed
in
inter-universal
Teichmüller
theory,
a
natural
functorial
isomorphism
class
of
equivalences
of
categories
—
cf.
the
discussion
of
“Monoids
and
Categories”
in
[IUTchI],
§0].
On
the
other
hand,
perhaps
the
most
fundamental
example,
in
the
context
of
inter-universal
Teichmüller
theory,
of
this
phenomenon
of
descent
is
the
following
[cf.
the
notational
conventions
of
[IUTchI],
Definition
3.1,
(e),
(f)]:
(MnDsc)
The
topological
multiplicative
monoid
determined
by
the
topological
ring
given
by
[the
union
with
{0}
of]
O
(Π
X
)
[cf.
[Alien],
Example
2.12.3,
(iii)]
—
that
is
to
say,
a
construction
that,
a
priori,
from
the
point
of
view
of
[AbsTopIII],
Theorem
1.9;
[AbsTopIII],
Corollary
1.10,
is
a
functorial
algorithm
in
the
topological
group
Π
X
[i.e.,
“Π
v
”,
from
the
point
of
view
(sQGOut)]
—
in
fact
descends
[cf.
the
discussion
at
the
beginning
of
[Alien],
§2.12;
the
discussion
of
[Alien],
Example
2.12.3,
(i)],
relative
to
passage
to
the
underlying
quotient
group
discussed
in
(SQGOut),
to
a
functorial
algorithm
in
the
topological
group
G
k
[i.e.,
“G
v
”,
from
the
point
of
view
(sQGOut)].
Finally,
we
remark
that
often,
in
inter-universal
Teichmüller
theory,
the
out-
put
data
of
the
functorial
algorithm
Φ
of
the
above
discussion
is
regarded
“stack-
theoretically”.
That
is
to
say,
the
output
data
is
not
a
single
“set-theoretic
object”,
but
rather
a
collection
[which
is
not
necessarily
a
set!]
of
set-theoretic
objects
linked
by
uniquely
determined
poly-isomorphisms
of
some
sort.
Typically,
this
sort
of
situation
arises
when
one
applies
(ExtInd2)
—
cf.
the
discussion
of
(NSsQ)
in
§3.8.
The
most
central
example
of
this
phenomenon
in
inter-universal
Teichmüller
theory
is
the
multiradial
algorithm
—
and,
especially,
the
portion
of
the
multi-
radial
algorithm
that
involves
the
log-Kummer-correspondence
and
closely
related
operations
of
Galois
evaluation
—
which
plays
the
role
of
exhibiting
the
Frobenius-like
Θ-pilot
as
one
possibility
within
a
collec-
tion
of
possibilities
constructed
via
anabelian
algorithms
from
étale-like
data
136
SHINICHI
MOCHIZUKI
[cf.
the
discussion
at
the
end
of
§3.6,
as
well
as
the
discussion
of
§3.10,
§3.11,
be-
low].
That
is
to
say,
the
log-Kummer-correspondence
and
closely
related
operations
of
Galois
evaluation
exhibit
the
Frobenius-like
Θ-pilot
as
one
possibility
within
a
collection
of
possibilities
constructed
via
anabelian
algorithms
from
étale-like
data
not
in
a
set-theoretic
sense
[i.e.,
one
possibility/element
contained
in
a
set
of
possibilities],
but
rather
in
a
“stack-theoretic
sense”,
in
accordance
with
various
applications
of
(ExtInd2)
[cf.
the
discussion
at
the
end
of
§3.6],
i.e.,
as
one
possibility,
up
to
isomorphism,
within
some
[not
necessarily
set-
theoretic!]
collection
of
possibilities.
As
discussed
in
(LVsQ)
in
§3.8,
one
arrives
at
a
set-theoretic
situation
—
i.e.,
one
possibility/element
contained
in
a
set
of
possibilities
—
only
after
one
obtains
a
“closed
loop”,
which
allows
one
to
pass
to
a
“coarse/set-theoretic
(sub)quotient”
by
taking
the
log-volume.
§3.10.
Detailed
description
of
the
chain
of
logical
AND
relations
We
begin
the
present
§3.10
with
the
following
well-known
and,
in
some
sense,
essentially
tautological
observation:
Just
as
every
form
of
data
—
i.e.,
ranging
from
text
files
and
webpages
to
audiovisual
data
—
that
can
be
processed
by
a
computer
can,
ultimately,
be
expressed
as
a
[perhaps
very
long!]
chain
of
“0’s”
and
“1’s”,
the
well-known
functional
completeness,
in
the
sense
of
propositional
calculus,
of
the
collection
of
Boolean
operators
consisting
of
logical
AND
“∧”,
logical
OR
“∨”,
and
negation
“¬”
motivates
the
point
of
view
that
one
can,
in
principle,
express
the
essential
logical
structure
of
any
mathematical
argument
or
theory
in
terms
of
elementary
logical
relations,
i.e.,
such
as
logical
AND
“∧”,
logical
OR
“∨”,
and
negation
“¬”.
Indeed,
it
is
precisely
this
point
of
view
that
formed
the
central
motivation
and
conceptual
starting
point
of
the
exposition
given
in
the
present
paper.
From
the
point
of
view
of
the
correspondence
with
the
terminology
and
modes
of
expression
that
actually
appear
in
[IUTchI-III]
and
[Alien],
the
representation
given
in
the
present
paper
of
the
essential
logical
structure
of
inter-universal
Te-
ichmüller
theory
in
terms
of
elementary
logical
relations,
i.e.,
such
as
logical
AND
“∧”
and
logical
OR
“∨”,
may
be
understood
as
follows:
·
Logical
AND
“∧”
corresponds
to
such
terms
as
·
simultaneous
execution
and
·
gluing
[cf.
[IUTchIII],
Remark
3.11.1,
(ii);
[IUTchIII],
Remark
3.12.2,
(ii),
(c
itw
),
(f
itw
);
the
final
portion
of
[Alien],
§3.7,
(i);
[Alien],
§3.11,
(iv)].
·
Logical
OR
“∨”
corresponds
to
such
terms
as
·
indeterminacies,
·
poly-morphisms,
and
·
projection/(sub)quotient/splitting
[cf.
§3.7;
the
title
of
[IUTchIII];
[IUTchIII],
Remark
3.9.5,
(xiii),
(ix);
[Alien],
§3.11,
(v);
[Alien],
§4.1,
(iv)].
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
137
Recall
the
essential
logical
structure
of
inter-universal
Teichmüller
theory
sum-
marized
in
(∧(∨)-Chn)
A
∧
B
=
A
∧
(B
1
∨
B
2
∨
.
.
.
)
=⇒
A
∧
(B
1
∨
B
2
∨
.
.
.
∨
B
1
∨
B
2
∨
.
.
.
)
=⇒
A
∧
(B
1
∨
B
2
∨
.
.
.
∨
B
1
∨
B
2
∨
.
.
.
∨
B
1
∨
B
2
∨
.
.
.
)
..
.
[cf.
the
discussion
of
§3.6,
§3.7].
Observe
that
if
the
description
of
the
various
“possibilities”
related
via
“∨’s”
in
the
above
displays
is
suitably
formulated,
i.e.,
without
superfluous
overlaps,
then
in
fact
these
logical
OR
“∨’s”
may
be
understood
˙
as
logical
XOR
“
∨’s”,
i.e.,
we
conclude
the
following:
˙
(∧(
∨)-Chn)
The
essential
logical
structure
of
inter-universal
Teichmüller
theory
may
be
summarized
as
follows:
A
∧
B
=⇒
A
∧
(B
1
∨
˙
B
2
∨
˙
.
.
.
)
A
∧
(B
1
∨
˙
B
2
∨
˙
.
.
.
∨
˙
B
1
∨
˙
B
2
∨
˙
.
.
.
)
=⇒
A
∧
(B
1
∨
˙
B
2
∨
˙
.
.
.
∨
˙
B
1
∨
˙
B
2
∨
˙
.
.
.
∨
˙
B
1
∨
˙
B
2
∨
˙
.
.
.
)
=
..
.
Here,
we
observe
the
following:
˙
(∧(
∨)-Chn1)
The
“∧’s”
in
the
above
display
·
arise
from
the
Θ-link,
which
may
be
thought
of
as
a
re-
lationship
between
certain
portions
of
the
multiplica-
tive
structures
of
the
ring
structures
arising
from
the
(Θ
±ell
NF-)Hodge
theaters
in
the
domain
and
codomain
of
the
Θ-link
that
are
common
[cf.
“∧”!]
to
these
ring
structures.
This
situation
is
reminiscent
of
·
the
fact
that
from
the
point
of
view
of
Boolean
alge-
bras,
“∧”
corresponds
to
the
multiplicative
structure
of
the
field
F
2
,
which
may
be
regarded,
via
the
splitting
determined
by
Teichmüller
representatives,
as
a
multi-
plicative
structure
that
is
common
[cf.
“∧”!]
to
Z
and
F
2
[cf.
Example
2.4.6,
(iii)],
as
well
as
of
·
the
discussion
of
[Alien],
§3.11,
(iv),
(2
and
),
concern-
ing
the
interpretation
of
the
discussion
of
crystals
in
[Alien],
§3.1,
(v),
(3
KS
),
in
terms
of
the
logical
rela-
tor
“∧”,
i.e.,
as
objects
that
may
be
simultaneously
interpreted,
up
to
isomorphism,
as
pull-backs
via
one
projection
morphism
and
[cf.
“∧”!]
as
pull-backs
via
the
other
projection
morphism.
138
SHINICHI
MOCHIZUKI
˙
˙
(∧(
∨)-Chn2)
The
“
∨’s”
in
the
above
display
may
be
understood
as
corre-
sponding
to
·
various
indeterminacies
that
arise
mainly
from
the
log-
Kummer-correspondence,
i.e.,
from
sequences
of
it-
erates
of
the
log-link,
which
may
be
thought
of
as
a
device
for
constructing
additive
log-shells.
The
addi-
tive
structures
of
the
ring
structures
arising
from
the
(Θ
±ell
NF-)Hodge
theaters
in
the
domain
and
codomain
of
the
Θ-link
are
structures
which,
unlike
the
corre-
sponding
multiplicative
structures,
are
not
common
˙
[cf.
“
∨”!]
to
these
ring
structures
in
the
domain
and
codomain
of
the
Θ-link.
This
situation
is
reminiscent
of
·
the
fact
that
from
the
point
of
view
of
Boolean
al-
˙
corresponds
to
the
additive
structure
of
gebras,
“
∨”
the
field
F
2
,
which
is
an
additive
structure
that
is
not
˙
shared
[cf.
“
∨”!],
relative
to
the
splitting
determined
by
Teichmüller
representatives,
by
Z
and
F
2
[cf.
Exam-
ple
2.4.6,
(iii)],
as
well
as
of
·
the
discussion
of
[Alien],
§3.11,
(iv),
(2
and
),
concerning
the
interpretation
of
the
discussion
of
crystals
in
[Alien],
§3.1,
(v),
(3
KS
)
in
terms
of
the
logical
relator
“∧”,
i.e.,
where
we
recall
that
the
two
pull-backs
of
the
rank
one
Hodge
subbundle
[cf.
[Alien],
§3.1,
(v),
(5
KS
);
the
discussion
of
Hodge
structures
in
[IUTchI],
§I2]
do
˙
not,
in
general,
coincide
[cf.
“
∨”!],
but
rather
differ
by
an
additive
“deformation
discrepancy”,
namely,
the
Kodaira-Spencer
morphism.
˙
˙
˙
(∧(
∨)-Chn3)
Taken
together,
(∧(
∨)-Chn1)
and
(∧(
∨)-Chn2)
may
be
under-
˙
stood
as
expressing
the
fact
that
the
“
∨’s”
and
“∧’s”
of
the
above
display
correspond,
respectively,
to
the
two
underlying
com-
binatorial
dimensions
—
i.e.,
addition
and
multiplication
—
of
a
ring
or,
alternatively,
to
the
two-dimensional
nature
of
the
log-theta-lattice
[cf.
the
discussion
of
[IUTchIII],
Remark
3.12.2,
(i);
the
latter
portion
of
[Alien],
§3.3,
(ii)].
Thus,
these
two
dimensions
may
be
understood,
alternatively,
as
correspond-
ing
to
·
the
arithmetically
intertwined
Θ-link
and
log-
link
of
inter-universal
Teichmüller
theory,
which
give
rise
to
the
multiradial
representation
up
to
suitable
˙
indeterminacies
[cf.
“∧(
∨)”!]
of
the
Θ-pilot;
·
the
description
given
in
Example
2.4.6,
(iii),
of
the
carry-addition
operation
on
the
truncated
ring
of
Witt
˙
[cf.
“∧(
∨)”!];
˙
vectors
Z/4Z
in
terms
of
“∧”
and
“
∨”
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
139
·
the
filtered
crystal
discussed
in
[Alien],
§3.1,
(v),
(5
KS
),
where
one
may
think
of
the
filtration
[i.e.,
rank
one
Hodge
subbundle]
as
“being
well-defined
up
to
inde-
˙
terminacies”
[cf.
“∧(
∨)”!],
i.e.,
up
to
a
“discrepancy”,
which
is
given
by
the
Kodaira-Spencer
morphism.
˙
˙
(∧(
∨)-Chn4)
The
two
dimensions
discussed
in
(∧(
∨)-Chn3)
may
be
under-
stood
as
corresponding
to
the
two
dimensions
—
i.e.,
·
the
successive
iterates
of
the
Frobenius
mor-
phism
in
positive
characteristic
and
·
successive
extensions
to
mixed
characteristic
—
of
a
ring
of
Witt
vectors
[cf.
the
discussion
of
the
latter
por-
tion
of
[Alien],
§3.3,
(ii)].
This
relationship
to
the
two
dimensions
of
a
ring
of
Witt
vectors
is
entirely
consistent
with
the
way
in
which
truncated
rings
of
Witt
vectors
occur
in
the
discussion
of
Example
2.4.6,
(iii),
i.e.,
with
the
expression
¨
=
(∧,
∨)
˙
∨
of
mixed
characteristic
“carry-addition”
as
a
sort
of
“inter-
twining”
between
addition
and
multiplication
in
the
field
F
2
obtained
by
“stacking”
multiplication
“∧”
on
top
of
addition
˙
“
∨”.
Moreover,
in
this
context,
we
note
that
the
various
correspondences
observed
in
˙
˙
(∧(
∨)-Chn3)
and
(∧(
∨)-Chn4)
are
particularly
fascinating
in
that
they
assert
that
the
“arithmetic
intertwining”
in
a
ring
between
addition
and
multiplication
—
i.e.,
the
mathematical
structure
which
is
in
some
sense
the
main
object
of
study
in
inter-universal
Teichmüller
theory
—
may
be
elucidated
by
means
of
a
the-
ory
[i.e.,
inter-universal
Teichmüller
theory!]
whose
essential
logical
structure,
˙
when
written
symbolically
in
terms
of
Boolean
operators
such
as
“∧”
and
“
∨”,
¨
=
∧
∨)
˙
in
Example
amounts
precisely
to
the
description
[cf.
the
discussion
of
(
∨
2.4.6,
(iii)]
of
the
“Boolean
intertwining”
that
appears
in
Boolean
carry-addition
¨
”
between
Boolean
addition
“
∨”
˙
and
Boolean
multiplication
“∧”.
Put
another
“
∨
way,
it
is
as
if
(TrHrc)
the
arithmetic
intertwining
which
is
the
main
object
of
study
in
inter-
universal
Teichmüller
theory
somehow
“induces”/“is
reflected
in”
a
sort
of
“structural
carry
operation”,
or
“trans-hierarchical
similitude”,
to
the
Boolean
intertwining
that
constitutes
the
essential
logical
struc-
ture
of
the
theory
[i.e.,
inter-universal
Teichmüller
theory]
that
is
used
to
describe
it:
arithmetic
intertwining
Boolean
intertwining!
Finally,
we
observe
that
it
is
also
interesting
to
note
that
the
essential
mechanism
underlying
the
Kummer
theory
of
theta
functions
—
which
plays
a
central
140
SHINICHI
MOCHIZUKI
role
in
inter-universal
Teichmüller
theory,
i.e.,
in
inducing
the
trans-hierarchical
similude
discussed
in
(TrHrc)
—
namely,
the
correspondence
Kummer
theory
of
theta
functions
←→
one
valuation/cusp
Kummer
theory
of
algebraic
rational
functions
←→
multiple
valuations/cusps
discussed
in
[IUTchIII],
Remark
2.3.3,
(viii),
(ix),
may
itself
be
thought
of
as
a
sort
of
trans-hierarchical
similitude
between
number
fields/local
fields
and
function
fields
over
number
fields/local
fields.
At
a
more
technical
level,
the
essential
logical
structure
of
inter-universal
Te-
˙
ichmüller
theory
summarized
symbolically
in
(∧(
∨)-Chn)
may
be
understood
as
consisting
of
the
following
steps:
(Stp1)
log-Kummer-correspondence
and
Galois
evaluation:
This
step
con-
sists
of
exhibiting
the
Frobenius-like
Θ-pilot
at
the
lattice
point
(0,
0)
of
the
log-theta-lattice
—
i.e.,
the
data
that
gives
rise
to
the
F
×μ
-prime-strip
that
appears
in
the
domain
of
the
Θ-link
—
as
one
possibility
within
a
collection
of
possibilities
[cf.
(ExtInd1)!]
constructed
via
anabelian
algorithms
from
holomor-
phic
[relative
to
the
0-column]
étale-like
data.
In
this
context,
it
is
perhaps
worth
mentioning
that
it
is
a
logical
tautology
that
the
content
of
the
above
display
may,
equivalently,
be
phrased
as
follows:
this
step
consists
of
the
negation
“¬”
of
the
assertion
of
the
non-existence
of
the
Frobenius-like
Θ-pilot
at
the
lattice
point
(0,
0)
of
the
log-
theta-lattice
—
i.e.,
the
data
that
gives
rise
to
the
F
×μ
-prime-
strip
that
appears
in
the
domain
of
the
Θ-link
—
within
the
col-
lection
of
possibilities
constructed
via
certain
anabelian
algo-
rithms
from
holomorphic
[relative
to
the
0-column]
étale-like
data
[cf.
also
the
discussion
of
(RcnLb)
below].
At
the
level
of
labels
of
lat-
tice
points
of
the
log-theta-lattice,
this
step
corresponds
to
the
descent
operation
(0,
0)
(0,
◦)
[cf.
the
discussion
at
the
end
of
§3.6;
the
discussion
at
the
end
of
§3.9;
[IUTchIII],
Remark
3.9.5,
(viii),
(sQ1),
(sQ2);
[IUTchIII],
Theorem
3.11,
(ii),
(iii)].
Finally,
we
recall
that
this
step
already
involves
the
introduc-
tion
of
the
(Ind3)
indeterminacy,
which
may
be
understood
as
a
sort
of
coarse
algorithmic
approximation
of
the
complicated
apparatus
constituted
by
the
log-Kummer-correspondence
and
Galois
evaluation
[cf.
(ExtInd1),
as
well
as
the
discussion
of
(logORInd)
in
§3.11
below;
the
discussion
of
the
algorithmic
parallel
transport
(APT)
property
in
[IUTchIII],
Remark
3.11.1,
(iv)].
(Stp2)
Introduction
of
(Ind1):
This
step
consists
of
observing
that
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
the
anabelian
construction
algorithms
of
(Stp1)
in
fact
descend
to
—
i.e.,
are
equivalent
to
algorithms
that
only
require
as
in-
put
data
the
weaker
data
constituted
by
[cf.
the
discussion
of
“descent”
in
§3.9]
—
the
associated
mono-analytic
étale-like
data,
i.e.,
in
the
notation
of
(sQGOut),
the
“G
v
’s”.
At
the
level
of
labels
of
lattice
points
of
the
log-theta-lattice,
this
step
corresponds
to
the
descent
operation
(0,
◦)
(0,
◦)
[cf.
[IUTchIII],
Remark
3.9.5,
(viii),
(sQ1),
(sQ2);
[IUTchIII],
Theorem
3.11,
(i),
as
well
as
the
references
to
[IUTchIII],
Theorem
3.11,
(i),
in
[IUTchIII],
Theorem
3.11,
(iii)].
Finally,
we
recall
that
this
step
involves
the
introduction
of
the
(Ind1)
indeterminacy,
which
[very
mildly!
—
cf.
the
discussion
of
(Ind3>1+2)
in
§3.11
below]
increases
the
collection
of
possibilities
under
consideration
[cf.
(ExtInd1)].
(Stp3)
Introduction
of
(Ind2):
This
step
consists
of
observing
that
the
anabelian
construction
algorithms
of
(Stp2)
in
fact
descend
to
—
i.e.,
are
equivalent
to
algorithms
that
only
require
as
in-
put
data
the
weaker
data
constituted
by
[cf.
the
discussion
of
“descent”
in
§3.9]
—
the
associated
mono-analytic
Frobenius-
like
data,
i.e.,
in
the
notation
of
(sQGOut)
and
(MnOut),
the
“G
v
O
F
×μ
’s”.
v
[That
is
to
say,
one
constructs
log-shells,
for
instance,
as
submonoids
of
“O
F
×μ
”,
as
opposed
to
subquotients
of
“G
v
”.]
At
the
level
of
labels
of
v
lattice
points
of
the
log-theta-lattice,
this
step
corresponds
to
the
descent
operation
(0,
◦)
(0,
0)
[cf.
[IUTchIII],
Remark
3.9.5,
(viii),
(sQ1),
(sQ2);
[IUTchIII],
Theorem
3.11,
(i),
as
well
as
the
references
to
[IUTchIII],
Theorem
3.11,
(i),
in
[IUTchIII],
Theorem
3.11,
(iii)].
Since
the
Θ-link
may
be
thought
of
as
a
sort
of
equivalence
of
labels
(0,
0)
⇐⇒
(1,
0)
—
i.e.,
corresponding
to
the
full
poly-isomorphism
of
F
×μ
-prime-strips
constituted
by
the
Θ-link
—
this
descent
operation
means
that
the
algo-
rithm
under
consideration
may
be
regarded
as
an
algorithm
whose
input
data
is
the
mono-analytic
Frobenius-like
data
(1,
0)
arising
from
the
codomain
of
the
Θ-link.
This
step
involves
the
introduction
of
the
(Ind2)
indeterminacy,
which
[very
mildly!
—
cf.
the
discussion
of
(Ind3>1+2)
in
§3.11
below]
increases,
at
least
from
an
a
priori
point
of
view,
the
collection
of
possibilities
under
consideration
[cf.
(ExtInd1)].
Finally,
we
recall
that
this
step
plays
the
important
role
of
141
142
SHINICHI
MOCHIZUKI
isolating
the
log-link
indeterminacies
in
the
domain
[i.e.,
the
(Ind3)
indeterminacy
of
(Stp1)]
and
the
codomain
[i.e.,
the
log-
shift
adjustment
discussed
in
(Stp7)
below]
of
the
Θ-link
from
one
another
[cf.
the
discussion
of
[IUTchIII],
Remark
3.9.5,
(vii),
(Ob7-2);
[Alien],
§3.6,
(iv)].
Here,
we
recall
[cf.
the
discussion
of
the
final
portion
of
[Alien],
§3.3,
(ii)]
that
these
log-link
indeterminacies
on
either
side
of
the
Θ-link
may
be
understood,
in
the
context
of
the
discussion
of
(InfH)
in
§3.3,
as
corresponding
to
the
copies
“C
×
”
on
either
side
of
the
double
coset
space
×
“C
×
\GL
+
2
(R)/C
”.
(Stp4)
Passage
to
the
holomorphic
hull:
The
passage
from
the
collection
of
possible
regions
that
appear
in
the
output
data
of
(Stp3)
to
the
collection
of
regions
contained
in
the
holomorphic
hull
—
relative
to
the
1-column
of
the
log-theta-lattice
—
of
the
union
of
possible
regions
of
the
output
data
of
(Stp3)
[cf.
[IUTchIII],
Remark
3.9.5,
(vi);
[IUTchIII],
Remark
3.9.5,
(vii),
(Ob5);
[IUTchIII],
Remark
3.9.5,
(viii),
(sQ3)]
is
a
simple,
straighforward
application
of
(ExtInd1),
that
is
to
say,
of
increasing
the
˙
set
of
possibilities
[i.e.,
of
“
∨’s”].
The
purpose
of
this
step,
together
with
(Stp5)
below,
is
to
pass
from
arbitrary
regions
to
regions
corresponding
to
arithmetic
vector
bundles
[cf.
[IUTchIII],
Remark
3.9.5,
(vii),
(Ob1),
(Ob2)].
(Stp5)
Passage
to
hull-approximants:
This
step
consists
of
passing
from
the
collection
of
arbitrary
regions
contained
in
the
holomorphic
hull
of
(Stp4)
to
hull-approximants,
i.e.,
regions
that
have
the
same
global
log-volume
as
the
original
“arbitrary
regions”,
but
which
correspond
to
arithmetic
vector
bundles
[cf.
[IUTchIII],
Remark
3.9.5,
(vii),
(Ob6);
[IUTchIII],
Remark
3.9.5,
(viii),
(sQ3)].
This
operation
does
not
affect
the
logical
“∧/∨”
structure
of
the
algorithm
since
this
operation
of
passing
to
hull-
approximants
does
not
affect
the
collection
of
possible
value
group
portions
—
i.e.,
“F
-prime-strips”
—
of
F
×μ
-prime-strips
determined
by
forming
the
log-volume
of
these
regions
[cf.
the
discussion
of
[IUTchIII],
Remark
2.4.2;
the
discussion
of
(IPL)
in
[IUTchIII],
Remark
3.11.1,
(iii)].
(Stp6)
Passage
to
a
suitable
positive
rational
tensor
power
of
the
de-
terminant:
This
step
consists
of
passing
from
the
[regions
corresponding
to]
arithmetic
vector
bundles
obtained
in
(Stp4),
(Stp5)
to
a
suitable
tensor
power
root
of
a
tensor
power
of
the
determinant
arithmetic
line
bundle
of
such
an
arithmetic
vector
bundle
[cf.
[IUTchIII],
Remark
3.9.5,
(vii),
(Ob3),
(Ob4);
[IUTchIII],
Remark
3.9.5,
(viii),
(sQ3)].
Just
as
in
the
case
of
(Stp5),
this
operation
does
not
affect
the
logical
“∧/∨”
structure
of
the
algorithm
since
this
operation
of
passing
to
a
suitable
positive
rational
tensor
power
of
the
determinant
does
not
affect
the
collection
of
possible
value
group
portions
—
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
i.e.,
“F
-prime-strips”
—
of
F
×μ
-prime-strips
determined
by
forming
the
log-volume
of
these
regions
[cf.
the
discussion
of
[IUTchIII],
Remark
2.4.2;
the
discussion
of
(IPL)
in
[IUTchIII],
Remark
3.11.1,
(iii)].
1-
column
..
.
1-
column
..
.
•
⏐
log
⏐
•
⏐
log
⏐
•
⏐
log
⏐
•
⏐
log
⏐
•
⏐
log
⏐
•
⏐
log
⏐
•
..
.
•
..
.
(Stp7)
Log-shift
adjustment:
The
arithmetic
line
bundles
that
appear
in
(Stp6)
occur
with
respect
to
the
arithmetic
holomorphic
structure
—
i.e.,
in
effect,
ring
structure
—
at
the
label
(1,
1)
of
the
log-theta-lattice,
i.e.,
at
a
label
vertically
shifted
by
+1
relative
to
the
label
(1,
0)
that
forms
the
codomain
of
the
Θ-link
[cf.
the
discussion
of
[IUTchIII],
Remark
3.9.5,
(vii),
(Ob8);
[IUTchIII],
Remark
3.9.5,
(viii),
(sQ4)].
That
is
to
say,
by
applying
the
algorithm
discussed
in
(Stp1)
∼
(Stp6)
at
each
lattice
point
(1,
m)
[where
m
∈
Z]
of
the
1-column
of
the
log-theta-lattice,
we
obtain
algorithms
with
input
data
at
(1,
m)
and
output
data
at
(1,
m
+
1)
—
cf.
the
diagonal
arrows
of
the
diagram
shown
above.
Thus,
the
totality
of
all
of
these
diagonal
arrows
may
be
thought
of
as
a
sort
of
endomorphism
of
the
1-column
of
the
log-theta-
lattice,
i.e.,
an
algorithm
whose
input
data
is
the
1-column
of
the
log-theta-lattice,
and
whose
output
data
lies
in
the
same
1-column
of
the
log-theta-lattice.
Indeed,
one
may
think
of
the
input
data
of
the
algorithm
discussed
in
(Stp1)
∼
(Stp6)
applied
at
the
lattice
point
(1,
m)
as
being
equipped
with
the
label
(1,
m)
,
while
the
output
data
of
this
algorithm
applied
at
the
lattice
point
(1,
m)
as
being
equipped
with
the
label
(1,
m
+
1),
i.e.,
where
one
regards
these
labels
(1,
m)
and
(1,
m
+
1)
as
being
“refinements”
of
the
respective
original
labels
(1,
m)
and
(1,
m
+
1)
of
the
1-column,
which
may
be
recovered
by
forgetting
the
additional
data
“”
constituted
by
the
input/output
of
the
algorithm
under
consideration.
Moreover,
one
may
143
144
SHINICHI
MOCHIZUKI
consider
a
“composite
refinement”
(1,
m)
,
i.e.,
obtained
by
taking
the
[q-
pilot
F
×μ
-prime-strip
with
the
same
label
as
the]
output
data
of
(1,
m)
as
the
input
data
of
(1,
m)
.
Here,
we
note
that
these
input/output
labels
“”
are,
in
effect,
implicit
in
the
species-theoretic
sense
[cf.
the
discussion
surrounding
(NSsQ)
in
§3.8;
the
discussion
of
[IUTchIV],
§3]
—
where
we
observe
that
the
“package
of
data”
constituted
by
a
species
may
be
understood
as
a
sort
of
label!
—
in
which
the
terms
“input
data”/“output
data”
are
used
throughout
[IUTchIII]
in
the
discussion
of
the
multiradial
algorithm
of
[IUTchIII],
Theorem
3.11.
Then
considering
the
totality
of
all
the
diagonal
arrows
corresponds
to
considering
the
“diagonal”
[i.e.,
in
the
sense
of
“symmetrized”]
collection
of
data,
relative
to
the
symmetry
given
by
Z
m
→
m
+
1
∈
Z,
which
induces
compatible
symmetries
(1,
m)
→
(1,
m
+
1)
;
(1,
m)
→
(1,
m
+
1)
[cf.
the
discussion
of
(Stp8)
below].
Here,
it
is
important
to
observe
that
the
use
of
these
labels
(1,
−)
,
(1,
−),
(1,
−)
renders
explicit
the
sense
in
which
the
gluing,
up
to
suitable
indeterminacies,
arising
from
the
[algorithm
denoted
by
the]
corresponding
diagonal
arrow
of
the
diagram
shown
above
between
the
input
data
(1,
m)
and
the
output
data
(1,
m
+
1)
is
—
not
(!)
a
gluing
embedded
in
some
familiar
ambient
space
[cf.
the
discusssion
of
(FxEuc),
(FxFld)
in
the
final
portion
of
§3.1],
but
rather
—
a
formal,
diagram-theoretic
gluing
between
data
with
distinct
labels
[i.e.,
induced,
in
effect,
by
the
labels
constituted
by
the
various
coordinates
of
the
log-theta-lattice,
via
the
various
descent
opera-
tions
that
appear
in
the
algorithm
discussed
in
(Stp1)
∼
(Stp6)],
hence,
in
particular,
does
not
give
rise
to
any
nontrivial
set-theoretic
conclusions
—
i.e.,
such
as
the
manifest
contradiction
(!)
that
arises,
if
one
arbitrarily
eliminates/forgets
the
input/output
labels
“”,
between
the
gluing
constituted
by
the
diagonal
arrows
of
the
diagram
shown
above
[i.e.,
between
the
local
value
group
portions
of
the
q-pilot
codomain
F
×μ
-prime-strips
and
domain
Θ-pilot
F
×μ
-prime-strips
of
the
arrows,
where
the
latter
is
subject
to
suitable
indeterminacies]
and
the
gluings
of
adjacent
local
value
groups/unit
groups
constituted
by
the
log-links
—
at
any
of
the
intermediate
steps
that
appear
in
the
course
of
the
execution
of
(Stp1)
∼
(Stp6).
That
is
to
say,
it
is
only
by
applying
the
symmetrization
procedure
de-
scribed
above
that
we
obtain
a
closed
loop
[cf.
the
discussion
of
Example
3.1.1,
(iii);
the
discussion
below
of
(DstMp),
(FxGl),
(NoCmpIss),
(En-
glf);
the
discussion
of
[IUTchIII],
Remark
3.9.5,
(ix);
[Alien],
§3.11,
(v)],
i.e.,
in
the
language
of
the
discussion
surrounding
(DltLb)
in
§3.11
below,
a
situation
that
simulates
—
via
the
introduction
of
suitable
indeter-
minacies
[cf.
the
discussion
of
(Stp8)
below]
—
a
situation
in
which
the
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
distinct
labels
on
the
domain
and
codomain
of
the
Θ-link
have
been
eliminated,
hence
allows
one
to
draw,
in
an
essentially
formal
manner
[cf.
the
discussion
of
(Stp8)
below],
nontrivial
set-theoretic
conclu-
sions
[cf.
the
discussion
surrounding
(NSsQ),
(LVsQ)
in
the
final
portion
of
§3.8].
Here,
we
note
that
the
diagonally
symmetrized
local
value
groups
that
one
must
consider
in
order
to
obtain
such
a
closed
loop
are,
in
effect,
obtained
by
pulling
back
these
local
value
groups
to
the
labels
“
(1,
m)
”
from
the
original
labels
“(1,
m)”
of
the
1-column
of
the
log-theta-lattice,
i.e.,
via
the
forgetting
operation
(1,
m)
(1,
m),
hence,
in
particular,
are
necessarily
subject
to
the
condition
of
compatibility
with
the
glu-
ings
of
adjacent
local
value
groups/unit
groups
constituted
by
the
log-links.
This
compatibility
is
established
by
passing
to
log-volumes
and
applying
the
invariance
of
the
log-volume
with
respect
to
the
log-link
[cf.
the
discussion
of
[IUTchIII],
Remark
3.9.5,
(vii),
(Ob9)],
which
may
be
inter-
preted
as
asserting,
in
effect,
that
the
log-link
induces,
via
passage
to
the
log-volume,
an
isomorphism
between
corresponding
adjacent
local
value
groups
in
the
1-column
of
the
log-theta-lattice
[cf.
(Stp8)
below].
Put
another
way,
this
compatibility
of
the
log-volume
with
the
log-links
in
the
1-column
of
the
log-theta-lattice
may
be
regarded
as
a
sort
of
“version/analogue
for
iterates
of
the
log-links
in
the
1-column”
of
the
saturation
with
respect
to
iterates
of
the
log-links
in
the
0-column
discussed
in
(logORInd)
[cf.
§3.11]
below,
hence,
in
particular,
as
a
sort
of
“1-column
version/analogue”
of
the
re-
markable
phenomenon
constituted
by
the
“stark
contrast
between
the
po-
tency
of
[the
0-column]
(logORInd)
and
the
utterly
meaningless
nature
of
(ΘORInd)”
[cf.
the
discussion
of
§3.11
below].
Finally,
in
this
context,
it
is
interesting
to
observe,
from
a
historical
point
of
view,
that
the
set-theoretic
confusion
[e.g.,
in
the
form
the
manifest
con-
tradiction
discussed
above!]
that
arises
at
intermediate
steps
in
the
course
of
the
execution
of
(Stp1)
∼
(Stp6)
if
one
does
not
take
into
account
the
various
labels
of
the
log-theta-lattice,
as
well
as
the
additional
input/output
labels
discussed
above
[i.e.,
which
are,
in
effect,
induced
by
various
labels
or
collections
of
labels
of
the
log-theta-lattice
],
is
remarkably
reminiscent
of
the
his-
torical
Weierstrass/Riemann
dispute
that
arose
in
complex
function
theory
in
the
context
of
the
theory
of
analytic
con-
tinuation/Riemann
surfaces,
i.e.,
prior
to
the
development
of
modern
axiomatic
set
theory
in
the
early
20-th
century
[cf.
the
discussion
of
§1.5]
—
where
we
note
that,
in
this
analogy,
the
various
labels
of
the
log-theta-lattice,
as
well
as
the
additional
input/output
labels
discussed
above,
correspond
to
the
set-theoretically
distinct
labels
of
various
copies
of
the
complex
open
unit
disc
that
appear
in
the
theory
of
analytic
continuation/Riemann
surfaces.
(Stp8)
Passage
to
log-volumes:
The
closed
loop
of
(Stp7)
[cf.
also
the
discussion
of
Example
3.1.1,
(iii);
the
discussion
below
of
(DstMp),
(FxGl),
(NoCmpIss),
(Englf)]
implies
that
the
crucial
logical
AND
“∧”
relation
145
146
SHINICHI
MOCHIZUKI
carefully
maintained
throughout
the
execution
of
(Stp1)
∼
(Stp7)
yields,
upon
taking
the
log-volume,
a
logical
AND
“∧”
relationship
between
the
original
q-pilot
input
F
-prime-strip
and
a
certain
algorithmically
constructed
collection
of
possible
output
F
-prime-strips
within
the
same
container,
i.e.,
some
copy
of
the
real
numbers
“R”
[cf.
[IUTchIII],
Remark
3.9.5,
(vii),
(Ob9);
[IUTchIII],
Remark
3.9.5,
(viii),
(sQ5);
[IUTchIII],
Remark
3.9.5,
(ix);
the
discussion
of
Substeps
(xi-d),
(xi-e)
of
the
proof
of
[IUTchIII],
Corollary
3.12;
the
discussion
of
[IUTchIII],
Remark
3.12.2,
(ii);
[Alien],
§3.11,
(v)].
The
inequality
in
the
statement
of
[IUTchIII],
Corollary
3.12,
then
follows
as
a
formal
consequence
of
the
invariance
of
the
log-volume
with
respect
to
the
log-
link
[cf.
the
discussion
of
the
final
portion
of
(Stp7)
above;
the
discussion
of
Substeps
(xi-e),
(xi-f),
(xi-g)
of
the
proof
of
[IUTchIII],
Corollary
3.12;
[IUTchIII],
Remark
3.12.2,
(iv),
(v)].
Here,
we
observe
that
the
various
indeterminacies
[such
as
(Ind1),
(Ind2),
(Ind3)]
that
arise
in
the
course
of
applying
(Stp1)
∼
(Stp7)
may
be
thought
of
as
a
sort
of
monodromy
associated
to
the
closed
loop
of
(Stp7)
[cf.
also
the
discussion
below
of
(DstMp),
(FxGl),
(NoCmpIss),
(Englf)].
In
this
context,
we
recall
from
(Stp7)
that
the
diagram
of
arrows
“
”
from
the
1-column
to
the
1-column
in
(Stp7)
admits
symmetries
(1,
m)
→
(1,
m
+
1)
[where
m
∈
Z]
that
are
compatible
with
all
of
the
arrows
in
the
diagram,
as
well
as
with
the
various
arrows
of
the
log-Kummer-correspondence
in
the
1-column.
These
symmetries
allow
one
to
synchronize
the
various
“monodromy
indeterminacies”
associated
to
each
“
”
[i.e.,
to
each
application
of
(Stp1)
∼
(Stp6)],
so
that
one
may
think
in
terms
of
a
single,
synchronized
collection
of
“monodromy
indeterminacies”
associated
to
the
totality
of
“
’s”
in
(Stp7).
Before
proceeding,
we
pause
to
examine
the
meaning
of
the
term
“closed
loop”
in
(Stp7),
(Stp8),
which
is
sometimes
a
source
of
confusion.
The
intended
meaning
of
this
term
is
that
the
sequence
of
mathematics
objects
on
which
the
series
of
operations
in
(Stp1)
∼
(Stp6)
[cf.
also
[IUTchIII],
Fig.
I.8]
are
performed
forms
a
closed
loop
in
the
sense
that
the
ultimate
output
data
lies
in
the
same
container
[i.e.,
up
to
a
log-shift
in
the
1-column,
as
discussed
in
(Stp7)]
as
the
input
data.
On
the
other
hand,
at
the
level
of
the
actual
mathematical
objects
that
one
is
working
with,
the
term
“closed
loop”
has
the
potential
to
result
in
certain
fundamental
misunderstandings,
since
it
may
be
[mistakenly!]
interpreted
as
suggesting
that
(DstMp)
one
is
considering
two
distinct
mappings
of
abstract
prime-strips
to
[Θ,
q-]pilot
prime-strips.
Once
one
takes
this
point
of
view
(DstMp),
there
is
inevitably
an
issue
of
diagram
commutativity,
i.e.,
the
issue
discussed
in
§3.6,
(Syp2),
that
one
must
contend
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
147
with.
As
discussed
in
Example
2.4.5,
(iv),
(v),
(vi),
(vii),
(viii)
[cf.
also
Examples
3.10.1,
3.10.2
below],
this
point
of
view
(DstMp)
corresponds
to
EssOR-IUT
[i.e.,
“essentially
OR
IUT”],
which,
as
the
name
suggests,
is
essentially
[thought
not
precisely!]
equivalent
to
OR-IUT,
and
in
particular,
constitutes
a
fundamental
misunderstanding
of
the
logical
structure
of
inter-universal
Teichmüller
theory.
Indeed,
the
chain
of
AND
relations
“∧”
discussed
in
§3.6,
as
well
as
the
present
§3.10,
which
lies
at
the
heart
of
the
essential
logical
structure
of
inter-
universal
Teichmüller
theory,
consists
precisely
of
(FxGl)
fixing
the
Frobenius-like
q-pilot
at
the
lattice
point
(1,
0),
as
well
as
the
gluing
[i.e.,
“∧”!]
via
the
Θ-link
of
this
q-pilot
at
(1,
0)
to
the
Frobenius-
like
Θ-pilot
at
the
lattice
point
(0,
0)
[cf.
[IUTchIII],
Remark
3.12.2,
(ii),
(c
itw
),
(e
itw
),
(f
itw
)].
One
then
proceeds
to
add
to
this
Θ-pilot
at
the
lattice
point
(0,
0)
more
and
more
˙
possibilities/indeterminacies
[i.e.,
“∨”,
or,
alternatively,
“
∨”!]
in
order
to
obtain
data
that
descends
to
the
same
label
[i.e.,
up
to
a
log-shift
in
the
1-column,
as
discussed
in
(Stp7)]
as
the
q-pilot
at
(1,
0).
That
is
to
say,
(NoCmpIss)
there
is
never
any
issue
of
compatibility
between
two
distinct
map-
pings
of
abstract
prime-strips,
as
in
(DstMp).
From
a
pictorial
point
of
view,
at
the
level
of
mathematical
objects,
one
is
working
in
(Stp1)
∼
(Stp8)
—
not
with
a
“closed
loop”
(!),
but
rather
—
a
single
fixed
line
segment
∧
•
=
=
=
=
=
•
corresponding
to
the
gluing
[i.e.,
“∧”!]
of
(FxGl)
[so
the
“•’s”
on
the
left
and
right
correspond,
respectively,
to
the
Θ-pilot
at
(0,
0)
and
the
q-pilot
at
(1,
0)],
which
˙
is
then
subjected
to
subsequent
“fuzzifications”
[i.e.,
“∨”,
or
alternatively,
“
∨”!]
of
the
Θ-pilot
at
(0,
0)
[denoted
in
the
following
display
by
the
notation
“(.
.
.
)”,
which
may
be
thought
of
as
representing
a
“fuzzy
disc”
that
contains
the
“•”
on
the
left]
=
=
=
∧
(
∨
•
=)
=
=
(
∨
∨
•
=
=)
=
(
∨
∨
∨
•
=
=
=)
(
∨
∨
∨
∨
•
=
=
=
∧
•
∧
∧
∧
∧
(
∨
∨
∨
∨
∨
•
=
=
=
(
∨
∨
∨
∨
∨
∨
•
=
=
=
∧
=
=
•
=
=
•
=
=
•
=
=
•
=)
=
•
=
=)
•
=
=
•
)
that
terminate
in
a
situation
[cf.
the
final
line
of
the
above
display]
in
which
(Englf)
the
fuzzifications
engulf
the
q-pilot
at
(1,
0),
i.e.,
a
situation
in
which
the
distinct
labels
may
be
eliminated
and
nontrivial
consequences
may
be
148
SHINICHI
MOCHIZUKI
obtained
[cf.
(Stp7),
(Stp8),
as
well
as
the
discussion
surrounding
(DltLb)
in
§3.11
below].
Thus,
in
summary,
throughout
the
series
of
fuzzification
operations
constituted
by
(Stp1)
∼
(Stp8),
the
line
segment
representing
the
gluing
[i.e.,
“∧”!]
of
(FxGl)
remains
fixed,
so
[cf.
(NoCmpIss),
(Englf)]
there
is
never
any
issue
of
compat-
ibility
between
two
distinct
mappings
of
abstract
prime-strips,
as
in
(DstMp).
Example
3.10.1:
Symmetries
as
a
fundamental
non-formal
aspect
of
glu-
ings.
One
psychological
aspect,
and
indeed
[in
many
cases]
possible
cause,
of
the
fundamental
misunderstandings
discussed
above
[cf.
the
discussion
above
sur-
rounding
(DstMp),
as
well
as
the
discussion
surrounding
(RfsDlg),
(DngPrc)
in
§1.12]
concerning
the
essential
logical
structure
of
inter-universal
Teichmüller
the-
ory
—
i.e.,
the
[erroneous!]
point
of
view
that
this
essential
logical
structure
of
inter-universal
Teichmüller
theory
should
be
understood
as
centering
around
an
issue
of
diagram
commutativity
—
is
the
following.
In
any
sort
of
gluing
sit-
uation
—
i.e.,
from
a
category-theoretic
point
of
view,
any
sort
of
situation
[cf.
the
numerous
examples
of
gluings
discussed
throughout
the
present
paper,
e.g.,
in
Examples
2.3.2,
2.4.1,
2.4.2,
2.4.3,
2.4.7,
2.4.8,
3.3.1,
3.5.2]
in
which
one
considers
the
[possibly
formal]
inductive
limit
I
of
a
diagram
of
the
form
Y
1
←−
X
−→
Y
2
—
there
is
a
certain
tendency
to
fall
into
the
“mental
trap”
of
believing
that
it
is
a
“tautology”
that
(UnvPrp)
any
conceivable
nontrivial/interesting
property
of
I
should
be
understood
in
terms
of
the
universal
property
of
an
inductive
limit,
i.e.,
the
property
to
the
effect
that
any
arrow
from
I
to
some
object
Z
should
be
understood
in
terms
of
the
issue
of
commutativity
of
a
diagram
of
the
form
X
−→
Y
2
⏐
⏐
⏐
⏐
?
Y
1
−→
Z
—
i.e.,
where
the
left-hand
vertical
and
upper
horizontal
arrows
are
the
arrows
in
the
definition
of
I.
Here,
we
observe
that
(UnvPrpOR)
the
diagram
commutativity
issue
that
arises
when
one
considers
the
uni-
versal
property
discussed
in
(UnvPrp)
is
essentially
a
logical
OR
“∨”
situation:
that
is
to
say,
one
can
pass
from
X
to
Z
in
the
diagram
of
(UnvPrp)
via
Y
1
OR
via
Y
2
,
but
there
is,
a
priori,
no
way
of
passing
from
such
an
“OR
situation”
to
the
desired
“AND
situation”
that
arises
when
commutativity
holds,
i.e.,
the
situation
where
one
knows
that
there
is
a
single
arrow
from
X
to
Z
that
is
simultaneously
equal
to
the
composite
of
the
two
arrows
that
pass
through
Y
1
and
equal
to
the
composite
of
the
two
arrows
that
pass
through
Y
2
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
149
—
cf.
the
discussion
of
Example
2.4.5,
(iv),
(v),
(vi),
(vii),
(viii).
Of
course,
it
is
indeed
a
tautology
that
I
satisfies
a
universal
property
as
in
(UnvPrp),
but
the
point
is
that
(FlsUnv)
the
fact
that
I
satisfies
such
a
universal
property
does
not
by
any
means
imply
—
i.e.,
as
one
might
falsely
conclude
from
the
nuance
carried
by
the
word
“universal”
in
ordinary,
non-technical
contexts!
—
that
any
con-
ceivable
nontrivial/interesting
property
of
I
is
best
understood
in
terms
of
this
universal
property.
That
is
to
say,
(SymPrp)
there
are
numerous
examples
in
mathematics
of
objects
“I”
that
are
constructed
as
gluings,
but
that
satisfy
important
nontrivial
properties
such
as
symmetry
properties
—
e.g.,
of
the
sort
discussed
in
§3.2
[cf.,
especially,
Example
3.2.2!]
—
that
do
not
admit
any
natural
“general
nonsense”
formulation
in
terms
of
the
universal
property
of
(UnvPrp).
In
the
spirit
of
(UnvPrpOR),
it
is
of
interest
to
note
that
(SymPrpAND)
symmetry
properties
typically
concern
some
sort
of
invariant,
or
“coric”,
structure
that
is
commonly
shared
throughout
various
“local-
izations”
of
the
diagram
that
gives
rise
to
I,
where
we
recall
that
it
is
es-
sentially
a
tautology
that
this
sort
of
notion
of
a
commonly
shared
“coric”
property
is
a
logical
AND
“∧”
situation
—
cf.
the
discussion
of
Example
2.4.5,
(iv),
(v),
(vi),
(vii),
(viii),
as
well
as
the
dis-
cussion
of
(FxGl),
(NoCmpIss),
(Englf),
above.
Well-known
elementary
examples
of
the
phenomenon
discussed
in
(SymPrp)
include
(i)
the
projective
general
linear
symmetries
[i.e.,
“P
GL
2
”]
of
the
pro-
jective
line,
which
may
be
constructed
as
a
gluing
of
two
copies
of
the
affine
line
[cf.
Example
2.4.7,
(iv),
(v)];
(ii)
the
group
structure
of
an
elliptic
curve,
which,
as
is
well-known,
may
be
constructed
as
a
gluing
of
two
cubic
planar
affine
curves;
(iii)
the
group
of
general
linear
oriented
symmetries
GL
+
2
(R)
of
the
two-
dimensional
R-vector
space
R
2
—
cf.
the
discussion
of
the
double
coset
space
×
C
×
\GL
+
2
(R)/C
in
§3.3,
(InfH)
—
in
the
situation
of
the
complex
Teichmüller
defor-
mations
discussed
in
Example
3.3.1,
where
we
recall
that
this
situation
may
be
regarded
[cf.
the
discussion
at
the
beginning
of
Example
3.5.2]
as
a
“gluing”
of
two
distinct
copies
of
the
complex
plane
C
along
a
common
underlying
two-dimensional
R-vector
space
R
2
[cf.
also
Example
3.5.2,
(iii)],
and
we
observe
that
these
symmetries
GL
+
2
(R)
allow
one
to
re-
gard
each
of
the
two
holomorphic
structures
[i.e.,
copies
of
C]
that
appear
in
this
gluing
as
indeterminate
GL
+
2
(R)-conjugates
of
the
subgroup
+
×
C
⊆
GL
2
(R)
inside
the
common
container
GL
+
2
(R);
(iv)
the
common
holomorphic
structure
that
appears
in
the
classical
theory
of
analytic
continuation
of
one-variable
complex
holomorphic
150
SHINICHI
MOCHIZUKI
functions
—
cf.,
e.g.,
the
discussion
of
the
theory
of
analytic
continuation
of
the
complex
logarithm
in
the
discussion
surrounding
(FxEuc)
in
§3.1,
as
well
as
in
the
historical
discussion
of
§1.5
and
the
discussion
of
the
toral
rotations
“C
×
”
in
§3.3,
(InfH)
—
where
we
observe
that
such
analytic
continuations
may
be
regarded
as
gluings
of
copies
of
the
complex
open
unit
disc,
and
that
the
common
holomorphic
structure
may
be
regarded
as
a
sort
of
common
symmetry
of
such
gluings,
i.e.,
if
one
thinks
in
terms
of
“almost
complex
structures”,
that
is
to
say,
in
terms
√
of
the
symmetry
of
the
tangent
bundle
given
by
multiplication
by
i
=
−1.
Here,
we
observe
in
passing
that
it
is
of
interest
to
note,
in
the
context
of
(ii),
that
the
group
structure
of
an
elliptic
curve,
as
well
as
the
existence
of
invariant
[i.e.,
with
respect
to
this
group
structure]
differentials
on
the
elliptic
curve,
play
a
central
role
in
the
argument
discussed
in
Example
3.2.1
—
an
argument
that
itself
played
a
fundamental
role
in
motivating
the
essential
logical
structure
of
inter-universal
Teichmüller
theory
[cf.
Example
3.2.1,
(viii)].
Of
course,
as
discussed
extensively
in
§3.3
[cf.
(InfH);
Examples
3.3.1,
3.3.2],
the
situations
discussed
in
(iii),
(iv)
also
exhibit
numerous
important
structural
similarities
to
various
important
aspects
of
inter-universal
Teichmüller
theory.
Example
3.10.2:
Chains
of
logical
AND
relations
via
commutative
dia-
grams.
We
maintain
the
notation
of
Example
3.10.1.
(i)
As
discussed
at
the
beginning
of
Example
3.10.1,
(ORAch)
the
“diagram-commutativity”,
or
“OR
approach”
[cf.
(UnvPrp),
(UnvPrpOR)],
X
−→
Y
2
Y
1
←−
?
Z
to
analyzing
the
structure
of
the
[possibly
formal]
inductive
limit
I
of
the
upper
line
of
the
above
diagram
constitutes
a
sort
of
“mental
trap”
that
appears
to
be,
in
many
cases,
one
of
the
main
causes
of
the
fundamental
misunderstandings
that
exist
in
certain
sectors
of
the
mathematical
community
concerning
inter-universal
Teichmüller
theory.
[Here,
we
recall
that,
in
the
situation
considered
in
inter-universal
Teichmüller
theory,
the
initial
gluing
in
the
first
horizontal
line
of
the
diagram
of
(ORAch)
corresponds
to
the
gluing
constituted
by
the
Θ-link
between
the
“Θ-pilot
object
in
the
Θ-(Θ
±ell
NF-)Hodge
theater”
Y
1
and
the
“q-pilot
object
in
the
q-(Θ
±ell
NF-
)Hodge
theater”
Y
2
along
the
prime-strip
data
X,
while
“Z”
is
to
be
understood
as
some
sort
of
container
that
contains
both
Y
1
and
Y
2
.]
The
state
of
affairs
summarized
in
(ORAch)
thus
prompts
the
following
question:
(Q∧(∨)CCD)
Since
the
tendency
of
many
mathematicians
—
especially
those
who
work
in
abstract
areas
of
arithmetic
geometry!
—
to
try
to
interpret
the
essential
logical
structure
of
inter-universal
Teichmüller
theory
in
terms
of
the
[incorrect!]
“OR
approach”
of
(ORAch)
appears
to
stem,
to
a
substantial
extent,
from
the
fact
that
such
mathematicians
often
have
a
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
151
strong
desire
to
formulate
structural
properties
of
mathematical
objects
in
terms
of
commutative
diagrams,
is
it
possible
to
somehow
formulate
the
essential
logical
structure
of
inter-universal
Teichmüller
theory
˙
summarized
in
(∧(∨)-Chn)
—
or,
alternatively,
(∧(
∨)-Chn)
—
in
terms
of
some
sort
of
collection
of
commutative
diagrams?
Here,
we
recall
from
the
discussion
at
the
beginning
of
the
present
§3.10
[cf.
also
the
discussion
surrounding
(SymIUT)
in
§1.12]
that
the
main
thrust
of
the
present
paper
lies
in
formulating
the
essential
logical
structure
of
inter-universal
Teichmüller
theory
[modulo
certain
“blackboxes”
in
anabelian
geometry
and
the
theory
of
étale
theta
functions]
in
terms
of
logical
AND
“∧”/logical
OR
“∨”
relations.
The
fundamental
motivation
for
this
approach
taken
in
the
present
paper
lies
in
the
point
of
view
that
such
logical
AND
“∧”/logical
OR
“∨”
relations
constitute
the
“most
primitive/fundamental/universal”
means
available
for
documenting
the
essential
logical
structure
of
a
mathematical
argument.
This
point
of
view
is
also
closely
related
to
the
point
of
view
of
computer
verification
of
mathematical
arguments
discussed
at
the
beginning
of
§1.12
[cf.
(CmbVer),
(Algor)].
On
the
other
hand,
even
if
this
point
of
view
is
in
some
sense
“correct”
from
an
abstract,
theoretical
standpoint,
it
is
not
necessarily
the
case
that
this
point
of
view
is
also
correct
from
the
somewhat
more
practical
standpoint
of
developing
an
optimally
efficient
means
of
communicating
the
essential
logical
structure
of
inter-universal
Teichmüller
theory
to
other
mathematicians.
It
is
precisely
this
practical
standpoint
that
motivates
the
question
posed
in
(Q∧(∨)CCD).
(ii)
In
a
word,
it
is
not
very
difficult
to
give
an
affirmative
answer
to
the
question
posed
in
(Q∧(∨)CCD).
Indeed,
(∧(∨)CCD)
one
may
formulate
the
essential
logical
structure
of
inter-universal
˙
Teichmüller
theory
summarized
in
(∧(∨)-Chn)
—
or,
alternatively,
(∧(
∨)-
Chn)
—
in
terms
of
a
chain
of
[tautologically!]
commutative
dia-
grams
as
follows:
Y
1
←−
X
−→
Y
2
⏐
⏐
!
!
!
!
Y
1
⏐
⏐
←−
X
−→
Y
2
!
!
!
Y
1
..
.
←−
X
..
.
−→
Y
2
..
.
!
—
where
the
left-hand
vertical
arrows
are
natural
inclusion
morphisms
into
larger
and
larger
containers,
and
“
!”
denotes
the
tautological
commutativity
of
the
square
in
question.
That
is
to
say,
this
chain
of
[tautologically!]
commutative
diagrams
exhibits
the
initial
gluing
in
the
first
horizontal
line
—
i.e.,
the
gluing
constituted
by
the
Θ-
link
between
the
“Θ-pilot
object
in
the
Θ-(Θ
±ell
NF-)Hodge
theater”
Y
1
and
the
“q-pilot
object
in
the
q-(Θ
±ell
NF-)Hodge
theater”
Y
2
along
the
prime-strip
data
X
—
as
an
object
that
maps
tautologically
to
the
gluings
in
the
subsequent
horizontal
lines,
which
consist
of
copies
of
the
initial
gluing
in
the
first
horizontal
line,
but
152
SHINICHI
MOCHIZUKI
with
the
“Θ-pilot
object
in
the
Θ-(Θ
±ell
NF-)Hodge
theater”
Y
1
regarded
up
to
suitable
indeterminacies,
or
“alternative
possibilities”
[i.e.,
as
discussed
in
(Stp1)
∼
(Stp8)],
exhibited
in
larger
and
larger
containers
“Y
1
”,
“Y
1
”,
.
.
.
.
Put
another
way,
this
approach
consists
of
embedding
Y
1
into
larger
and
larger
containers
[i.e.,
the
containers
obtaining
by
adding
the
various
indeterminacies]
until
the
container
becomes
sufficiently
large
as
to
contain/engulf
[not
only
Y
1
,
but
also
(!)]
Y
2
[cf.
the
discussion
surrounding
(Englf)].
(iii)
At
first
glance,
the
approach
discussed
in
(∧(∨)CCD)
may
appear
to
some
arithmetic
geometers
to
be
entirely
unfamiliar
and
fundamentally
different
from
the
approach
taken
in
the
numerous
theories
in
arithmetic
geometry
that
existed
prior
to
the
appearance
of
inter-universal
Teichmüller
theory.
In
fact,
however,
the
approach
discussed
in
(∧(∨)CCD)
is
entirely
analogous
to
the
approach
taken
in
the
classical
theory
of
crystals.
Moreover,
this
analogy
with
the
classical
the-
ory
of
crystals
is
completely
compatible
with
the
discussion
of
the
strong
structural
resemblances
between
inter-universal
Teichmüller
theory
and
the
theory
of
crys-
tals
given
in
[Alien],
§3.1,
(v)
[cf.
also
§3.5
of
the
present
paper,
as
well
as
the
˙
discussion
of
(∧(
∨)-Chn)
in
the
present
§3.10].
Indeed,
the
“container
of
inde-
terminacies”
discussed
in
(ii)
may
be
understood
as
corresponding
to
the
PD-
envelopes/thickenings
that
apppear
in
the
theory
of
crystals.
That
is
to
say,
the
approach
typically
taken
in
the
theory
of
crystals
to
constructing
[nontrivial!]
crystals
—
i.e.,
to
constructing
isomorphisms
∼
p
∗
1
(−)
→
p
∗
2
(−)
between
the
pull-backs
via
the
two
natural
projections
Y
p
1
←−
Y
×
Y
p
2
−→
Y
of
some
object
“(−)”
on
some
scheme
Y
—
proceeds
not
by
securing
some
sort
of
commutative
diagram
p
1
p
2
Y
←−
Y
×
Y
−→
Y
?
Z
—
i.e.,
corresponding
to
the
[incorrect!]
“OR
approach”
of
(ORAch)!
—
such
that
the
object
“(−)”
on
Y
descends
to
Z,
but
rather
by
restricting
to
the
“suf-
ficiently
large
container”
constituted
by
a
suitable
PD-envelope/thickening
of
the
diagonal
in
Y
×
Y
and
verifying
that
this
container
is
indeed
sufficiently
large
that
the
restriction
to
this
PD-envelope/thickening
of
p
∗
1
(−)
already
contains,
up
to
isomorphism,
the
inverse
image
p
−1
2
(−).
§3.11.
The
central
importance
of
the
log-Kummer-correspondence
In
the
context
of
the
discussion
of
§3.10,
it
is
important
to
recall
that,
whereas
(Stp2)
∼
(Stp8)
are
technically
trivial
in
the
sense
that
they
concern
operations
that
are
very
elementary
and
only
require
a
few
lines
to
describe,
the
log-Kummer-
correspondence
and
Galois
evaluation
operations
that
comprise
(Stp1)
depend
on
the
highly
nontrivial
theory
of
[EtTh]
and
[AbsTopIII].
Moreover,
the
technical
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
153
description
of
these
operations
that
comprise
(Stp1)
occupies
the
bulk
of
[IUTchI-
III].
The
central
importance
of
(Stp1)
may
also
be
seen
in
the
subordinate
nature
of
(Ind1),
(Ind2)
[which
occur
in
(Stp2),
(Stp3)]
relative
to
(Ind3)
[which
occurs
in
(Stp1)],
i.e.,
in
the
sense
that
(Ind3>1+2)
once
one
constructs
the
output
of
the
multiradial
representation
of
the
Θ-pilot
[cf.
[IUTchIII],
Theorem
3.11,
(ii)]
via
tensor-packets
of
log-
shells
in
such
a
way
that
each
local
portion
of
this
output
is
stable
with
respect
to
the
indeterminacy
(Ind3),
these
local
portions
of
the
output
are
automatically
“essentially
stable”
[i.e.,
stable
up
to
discrepancies
at
the
valuations
∈
V
bad
that
affect
the
resulting
log-volumes
only
up
to
very
small/essentially
negligible
order]
with
respect
to
the
indeterminacies
(Ind1),
(Ind2)
[cf.
[IUTchIII],
Theorem
3.11,
(i)].
Finally,
we
observe
that
this
property
(Ind3>1+2)
is
strongly
reminiscent
of
the
discussion
of
(CnfInd1+2)
and
(CnfInd3)
in
§3.5.
One
way
to
understand
the
content
of
the
operations
of
(Stp1)
is
as
follows.
These
operations
may
be
regarded
as
a
sort
of
(logORInd)
saturation
of
the
Frobenius-like
Θ-pilot
at
the
lattice
point
(0,
0)
of
the
log-theta-lattice
—
i.e.,
which
is
linked,
via
the
Θ-link,
to
the
Frobenius-like
q-pilot
at
(0,
1)
—
with
respect
to
all
of
the
possibil-
ities
that
occur
in
the
0-column
of
the
log-theta-lattice,
i.e.,
all
of
the
possibilities
that
arise
from
a
possible
confusion
between
the
domain
and
codomain
of
the
log-links
in
the
0-column
[cf.
the
description
of
(Stp1)].
In
this
sense,
the
content
of
(Stp1)
is
formally
reminiscent
of
the
“(NeuORInd)”
that
appeared
in
the
discussion
of
§3.4,
i.e.,
which
may
be
understood
as
a
sort
of
(ΘORInd)
saturation
of
the
Frobenius-like
Θ-pilot
at
the
lattice
point
(0,
0)
of
the
log-theta-lattice
with
respect
to
all
of
the
possibilities
—
i.e.,
“Θ-plt”,
“q-plt”
[cf.
(NeuORInd2)]
—
that
arise
from
a
possible
confusion
between
the
domain
and
codomain
of
the
Θ-link
joining
the
lattice
points
(0,
0)
and
(0,
1).
[In
this
context,
we
note
that
the
logical
OR
“∨’s”
that
appear
in
(logORInd),
˙
(ΘORInd)
may
in
fact
be
understood
as
logical
XOR
“
∨’s”
—
cf.
the
discussion
˙
surrounding
(∧(
∨)-Chn)
in
§3.10.]
On
the
other
hand,
whereas,
as
observed
in
the
discussion
at
the
end
of
§3.4,
(ΘORInd)
yields
a
meaningless/useless
situation
that
does
not
give
rise
to
any
interesting
mathematical
consequences,
(logORInd),
by
contrast,
is
a
highly
potent
technical
device
that
forms
the
technical
core
of
inter-universal
Teichmüller
theory.
Before
preceding,
we
observe
that,
in
this
context,
it
is
interesting
to
note
that
both
of
these
“saturation
operations”
(logORInd)
and
(ΘORInd)
are
in
some
sense
qualitatively
similar
to
the
label
crushing
operation
(ExtInd2).
Indeed,
(ExtInd2)
consists,
roughly
speaking,
of
regarding
mathematical
objects
of
a
certain
type
up
to
isomorphism,
i.e.,
of
saturating
within
an
isomorphism
class
of
mathematical
objects
of
a
certain
type
[cf.
the
discussion
of
(ExtInd2)
in
§3.8,
as
well
as
the
discussion
of
(DltLb)
below].
154
SHINICHI
MOCHIZUKI
The
stark
contrast
between
the
potency
of
(logORInd)
and
the
utterly
mean-
ingless
nature
of
(ΘORInd)
is
highly
reminiscent
of
the
central
role
played,
in
Example
3.3.2,
(iv),
by
invariance
with
respect
to
ι
=
01
−1
0
∈
C
×
⊆
GL
+
2
(R)
[where
we
recall
from
(InfH)
that
C
×
corresponds
to
the
log-link!],
which
lies
in
stark
contrast
to
the
utterly
meaningless
nature
of
considering
invariance
with
respect
to
dilations
λ
0
1
0
∈
GL
+
2
(R)
[where
we
recall
from
(InfH)
that
such
dilations
correspond
to
the
Θ-link].
One
way
to
witness
the
potency
of
(logORInd)
is
as
follows.
Recall
that
the
Θ-link,
by
definition
[cf.
[IUTchIII],
Definition
3.8,
(ii)],
consists
of
·
a
dilation
applied
to
the
local
value
group
portions
of
the
ring
structures
in
its
domain
and
codomain,
coupled
with
·
a
full
poly-isomorphism
—
which
preserves
log-volumes,
hence
is
non-
dilating!
—
between
the
local
“O
×μ
’s”,
i.e.,
the
local
unit
group
portions,
of
these
ring
structures.
By
contrast,
the
log-links
in
the
0-column
of
the
log-theta-lattice
have
the
effect
of
“juggling/rotating/permuting”
the
local
value
group
portions
and
local
unit
group
portions
of
the
ring
structures
that
appear
in
this
0-column
[cf.,
e.g.,
the
discussion
of
[Alien],
Example
2.12.3,
(v)].
From
this
point
of
view,
the
tautologically
vertically
coric
—
i.e.,
invariant
with
respect
to
the
application
of
the
log-link!
—
nature
of
the
output
data
of
(logORInd)
is
already
somewhat
“shocking”
in
nature.
That
is
to
say,
the
tautologically
vertically
coric
nature
of
this
output
data
of
(logORInd)
suggests
that
(Di/NDi)
this
output
data
already
exhibits
some
sort
of
equivalence,
up
to
per-
haps
some
sort
of
mild
discrepancy,
between
the
dilated
and
non-dilated
portions
of
the
Θ-link.
Such
an
equivalence
already
strongly
suggests
that
some
sort
of
bound
on
heights
should
follow
as
a
formal
consequence,
i.e.,
in
the
style
of
the
classical
argument
that
implies
the
isogeny
invariance
of
heights
of
elliptic
curves
[cf.
the
discussion
of
[Alien],
§2.3,
§2.4,
as
well
as
the
discussion
of
Example
3.2.1
in
the
present
paper;
the
discussion
of
§3.5
in
the
present
paper].
Finally,
we
conclude
by
emphasizing
that,
in
inter-universal
Teichmüller
theory,
(DltLb)
ultimately
one
does
want
to
find
some
way
in
which
to
delete/eliminate
the
distinct
labels
on
the
Θ-
and
q-pilot
objects
[i.e.,
“Θ-plt”
and
“q-plt”]
in
the
domain
and
codomain
of
the
Θ-link
[cf.
the
discussion
of
Example
3.1.1,
(iii);
the
discussion
of
(AOL4),
(AOΘ4)
in
§3.4;
(Stp7),
(Stp8)
in
§3.10],
that
is
to
say,
not
via
the
naive,
simple-minded
approach
of
(ΘORInd)
[i.e.,
(NeuORInd2)
in
the
discussion
of
§3.4],
but
rather
via
the
indirect
approach
of
applying
descent
operations
(0,
0)
(Stp1)
(0,
◦)
(Stp2)
(0,
◦)
(Stp3)
(0,
0)
(Stp3)
⇐⇒
(1,
0)
LOGICAL
STRUCTURE
OF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
155
as
discussed
in
(Stp1)
∼
(Stp8)
[cf.,
especially,
(Stp7),
(Stp8)]
of
§3.10,
i.e.,
an
approach
that
centers
around
(logORInd).
This
approach
is
based
on
the
vari-
ous
anabelian
reconstruction
algorithms
discussed
in
(Stp1)
∼
(Stp3),
which
allow
one
to
exhibit
the
Frobenius-like
Θ-pilot
object
at
(0,
0)
as
one
possibil-
ity
among
some
broader
collection
of
possibilities
that
arise
from
the
introduction
of
various
types
of
indeterminacy.
In
this
context,
we
observe
[cf.
the
discus-
sion
of
(ExtInd2),
(NSsQ)
at
the
end
of
§3.9]
that
since
such
anabelian
recon-
struction
algorithms
only
reconstruct
various
types
of
mathematical
objects
[i.e.,
monoids/pseudo-monoids/mono-theta
environments,
etc.]
not
“set-theoretically
on
the
nose”
[i.e.,
not
in
the
sense
of
strict
set-theoretic
equality],
but
rather
up
to
[a
typically
essentially
unique,
if
one
allows
for
suitable
indeterminacies]
isomorphism,
it
is
not
immediately
clear
(RcnLb)
in
what
sense
such
anabelian
reconstruction
algorithms
yield
a
recon-
struction
of
the
crucial
labels
—
i.e.,
such
as
“(0,
0)”
—
that
underlie
the
crucial
logical
AND
“∧”
structure
discussed
in
§3.4
[cf.,
especially,
(AOL1),
(AOΘ1)].
The
point
here
is
that
indeed
such
anabelian
reconstruction
algorithms
are
not
capable
of
reconstructing
such
labels
“set-theoretically
on
the
nose”.
On
the
other
hand,
in
this
context,
it
is
important
to
recall
the
essential
substantive
content
of
the
various
labels
involved:
(HolFrLb)
(0,
0):
The
holomorphic
Frobenius-like
data
labeled
by
(0,
0)
con-
sists
of
various
monoids/pseudo-monoids/mono-theta
environments,
etc.,
regarded
as
abstract
monoids/pseudo-monoids/mono-theta
environments,
etc.,
i.e.,
as
objects
that
are
not
equipped
with
the
auxiliary
data
of
how
they
might
have
been
reconstructed
via
anabelian
algorithms
from
holo-
morphic
étale-like
data
labeled
(0,
◦)
[cf.
the
discussion
of
(UdOut),
(In-
Out),
(PSOut),
(ItwOut)
in
§3.9].
In
particular,
such
monoids/pseudo-
monoids/mono-theta
environments,
etc.,
are
not
invariant
with
respect
to
the
“juggling/rotating/permuting”
of
local
value
group
portions
and
lo-
cal
unit
group
portions
effected
by
the
log-links
in
the
0-column
of
the
log-theta-lattice,
but
rather
correspond
to
a
temporary
cessation
[cf.
the
label
(0,
0)
as
opposed
to
the
label
(0,
◦)!]
of
this
operation
of
jug-
gling/rotation/permutation.
(MnAlyLb)
(0,
0)
:
The
mono-analytic
Frobenius-like
data
labeled
by
(0,
0)
consists
of
the
F
×μ
-prime-strip
determined
by
the
Frobenius-like
Θ-
pilot
at
(0,
0),
regarded
as
an
abstract
F
×μ
-prime-strip
[cf.
the
discus-
sion
of
(UdOut),
(InOut),
(PSOut),
(ItwOut)
in
§3.9].
Thus,
the
transition
of
labels
(0,
0)
(0,
0)
consists
of
an
operation
of
forgetting
some
sort
of
auxiliary
structure
[cf.
the
discussion
of
(UdOut)
in
§3.9].
Here,
we
recall
that
this
construction
of
the
F
×μ
-prime-strip
determined
by
the
Frobenius-like
Θ-pilot
at
(0,
0)
is
technically
possible
precisely
because
of
the
“temporary
cessation”
discussed
above
[cf.
the
discussion
of
the
definition
of
the
Θ-link
in
[Alien],
§3.3,
(ii),
as
well
as
in
§3.3
of
the
present
paper].
156
SHINICHI
MOCHIZUKI
Thus,
the
nontrivial
substantive
content
of
the
anabelian
reconstruction
algorithms
of
(Stp1)
∼
(Stp3)
—
and
hence
of
the
descent
operations
(0,
0)
(Stp1)
∼
(Stp3)
(0,
0)
that
result
from
these
anabelian
reconstruction
algorithms
—
consists
of
statements
to
the
effect
that
(FrgInv)
the
operation
of
forgetting
discussed
in
(MnAlyLb)
can
in
fact,
if
one
allows
for
suitable
indeterminacies,
be
inverted.
It
is
precisely
this
invertibility
(FrgInv),
up
to
suitable
indeterminacies,
of
the
operation
of
forgetting
discussed
in
(MnAlyLb),
together
with
the
fact
that
(GluDt)
the
only
data
appearing
in
the
reconstruction
algorithms
[i.e.,
in
the
0-column]
that
is
glued
[cf.
the
discussion
of
[IUTchIII],
Remark
3.11.1,
(ii);
the
final
portion
of
[Alien],
§3.7,
(i),
as
well
as
the
discussion
of
the
(Ind2)
indeterminacy
in
the
final
portion
of
§3.11,
(Stp3),
in
the
present
paper]
to
data
in
the
1-column
is
the
F
×μ
-prime-strip
labeled
(0,
0)
,
that
ensures
that
the
descent
operations
discussed
above
do
indeed
preserve
the
crucial
logical
AND
“∧”
relations
discussed
in
§3.4,
§3.6,
§3.7,
§3.10,
i.e.,
even
though
the
reconstruction
algorithms
underlying
these
descent
operations
do
not
yield
reconstructions
of
the
various
labels
“(0,
0)”,
etc.,
“set-theoretically
on
the
nose”.
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